BR Maths Corner-1

[Amber G.]

2¹³⁶²⁷⁹⁸⁴¹−1, discovered recently, is the largest known prime.

It's a Mersenne prime (2ᵖ-1), which are easier to find. see my posts on Mersenne primes and how they are discovered in this dhaga.

( High-school students can understand the basics --and I have encouraged GIMPS to get students excited about mathematics. Students can run the free software are get interested in mathematical research)

It took nearly 6 years for the GIMPS software to find it after the previous largest known prime. It was also the first Mersenne prime found using GPUs.

--- Added later
Here is my post about 6 years about last such prime was discovered... after that exponents of 82589933 adnd recently 136279841. was discovered.

Reproducing that post:

In this dhaga I have talked about Mersenne prime numbers dozens of times. Now we have 50th known Mersenne prime and 50th perfect number!
New Mersenne prime M77232917 found! 2^77232917-1 is the largest known prime number!
https://www.mersenne.org/primes/press/M77232917.html

RALEIGH, NC., January 3, 2018 -- The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 277,232,917-1, having 23,249,425 digits. A computer volunteered by Jonathan Pace made the find on December 26, 2017. Jonathan is one of thousands of volunteers using free GIMPS software available at www.mersenne.org/download/.

The new prime number, also known as M77232917, is calculated by multiplying together 77,232,917 twos, and then subtracting one. It is nearly one million digits larger than the previous record prime number, in a special class of extremely rare prime numbers known as Mersenne primes. It is only the 50th known Mersenne prime ever discovered, each increasingly difficult to find. Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers more than 350 years ago. GIMPS, founded in 1996, has discovered the last 16 Mersenne primes. Volunteers download a free program to search for these primes, with a cash award offered to anyone lucky enough to find a new prime. Prof. Chris Caldwell maintains an authoritative web site on the largest known primes, and has an excellent history of Mersenne primes.

<snip ,, see the original post if interested>
For More Information on Mersenne Primes

Prime numbers have long fascinated both amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etc. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime number of the form 2P-1. The first Mersenne primes are 3, 7, 31, and 127 corresponding to P = 2, 3, 5, and 7 respectively. There are now 50 known Mersenne primes. { 2024 - now there are 52 known}

Mersenne primes have been central to number theory since they were first discussed by Euclid about 350 BC. The man whose name they now bear, the French monk Marin Mersenne (1588-1648), made a famous conjecture on which values of P would yield a prime. It took 300 years and several important discoveries in mathematics to settle his conjecture.

At present there are few practical uses for this new large prime, prompting some to ask "why search for these large primes"? Those same doubts existed a few decades ago until important cryptography algorithms were developed based on prime numbers. For seven more good reasons to search for large prime numbers, see here.

Previous GIMPS Mersenne prime discoveries were made by members in various countries.
In January 2016, Curtis Cooper et al. discovered the 49th known Mersenne prime in the U.S.
In January 2013, Curtis Cooper et al. discovered the 48th known Mersenne prime in the U.S.
In April 2009, Odd Magnar Strindmo et al. discovered the 47th known Mersenne prime in Norway.
In September 2008, Hans-Michael Elvenich et al. discovered the 46th known Mersenne prime in Germany.
In August 2008, Edson Smith et al. discovered the 45th known Mersenne prime in the U.S.
In September 2006, Curtis Cooper and Steven Boone et al. discovered the 44th known Mersenne prime in the U.S.
In December 2005, Curtis Cooper and Steven Boone et al. discovered the 43rd known Mersenne prime in the U.S.
In February 2005, Dr. Martin Nowak et al. discovered the 42nd known Mersenne prime in Germany.
In May 2004, Josh Findley et al. discovered the 41st known Mersenne prime in the U.S.
In November 2003, Michael Shafer et al. discovered the 40th known Mersenne prime in the U.S.
In November 2001, Michael Cameron et al. discovered the 39th Mersenne prime in Canada.
In June 1999, Nayan Hajratwala et al. discovered the 38th Mersenne prime in the U.S.
In January 1998, Roland Clarkson et al. discovered the 37th Mersenne prime in the U.S.
In August 1997, Gordon Spence et al. discovered the 36th Mersenne prime in the U.K.
In November 1996, Joel Armengaud et al. discovered the 35th Mersenne prime in France.

Euclid proved that every Mersenne prime generates a perfect number. A perfect number is one whose proper divisors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3 and the second perfect number is 28 = 1 + 2 + 4 + 7 + 14. Euler (1707-1783) proved that all even perfect numbers come from Mersenne primes. The newly discovered perfect number is 277,232,916 x (277,232,917-1). This number is over 46 million digits long! It is still unknown if any odd perfect numbers exist.

There is a unique history to the arithmetic algorithms underlying the GIMPS project. The programs that found the recent big Mersenne primes are based on a special algorithm. In the early 1990's, the late Richard Crandall, Apple Distinguished Scientist, discovered ways to double the speed of what are called convolutions -- essentially big multiplication operations. The method is applicable not only to prime searching but other aspects of computation. During that work he also patented the Fast Elliptic Encryption system, now owned by Apple Computer, which uses Mersenne primes to quickly encrypt and decrypt messages. George Woltman implemented Crandall's algorithm in assembly language, thereby producing a prime-search program of unprecedented efficiency, and that work led to the successful GIMPS project.

School teachers from elementary through high-school grades have used GIMPS to get their students excited about mathematics. Students who run the free software are contributing to mathematical research. David Stanfill's and Ernst Mayer's verification computations for this discovery was donated by Squirrels LLC (http://www.airsquirrels.com) which services K-12 education and other customers.

After that post, we have 51 and now 52nd such number!
Note that each such prime also means new 'prefect number' !

[Amber G.]

I have covered news and problems of many International Mathematical Olympiads here in this dhaga And encouraged and mentored many young minds ..

Glad to see that now there e is Special program for students who qualify or do good for olympiad to get direct admission to IIT. (Even without JEE).

We have announced a new admission pathway for our BTech and BS programs, based on performance in International Olympiads. Starting with the academic year 2025-26, this initiative seeks to recognize exceptionally talented students who have excelled in highly competitive international Olympiads in subjects such as Mathematics, Physics, Chemistry, Biology, and Informatics.

Under this program, IIT Kanpur will offer reserved seats in five departments: Biological Sciences & Bioengineering, Computer Science & Engineering, Chemistry, Economic Sciences, and Mathematics & Statistics. Admission through this route will allow a limited number of seats per department specifically for students who have undergone rigorous preparation and training in relevant Olympiad camps. This initiative offers a unique opportunity for those students who wish to pursue their passion in these areas at one of India’s leading institutions, bypassing the traditional JEE (Advanced) route.

Kudos to IIT Director.

Link: https://iitk.ac.in/doaa/data/Notificati ... mpiads.pdf

[sudarshan]

2¹³⁶²⁷⁹⁸⁴¹−1, discovered recently, is the largest known prime.

...

Nice post, but....

The newly discovered perfect number is 277,232,916 x (277,232,917-1). This number is over 46 million digits long! It is still unknown if any odd perfect numbers exist.

46 million digits? The number quoted above is 17 digits long. Maybe it meant the prime itself is 46 million digits long?

[Amber G.]

2¹³⁶²⁷⁹⁸⁴¹−1, discovered recently, is the largest known prime.

...

Nice post, but....

The newly discovered perfect number is 277,232,916 x (277,232,917-1). This number is over 46 million digits long! It is still unknown if any odd perfect numbers exist.

46 million digits? The number quoted above is 17 digits long. Maybe it meant the prime itself is 46 million digits long?

Thanks for the post. To make it clear :

277,232,917 is a prime number. It is 9 digits.

M77232917 = 2⁷⁷²³²⁹¹⁷−1 = (is calculated by multiplying together 77,232,917 twos, and then subtracting one) is a prime number discovered in 2018 . It is 23,249,425 digits long.

---- That perfect number , I was talking about, which is over 46 million digits long! is associated with M77232917 .

it is (2⁷⁷²³²⁹¹⁶)*( 2⁷⁷²³²⁹¹⁷−1)

---
(2^136279841-1) is the latests Mersenne prime it is 41,024,320 digits, At present there are 52 such know primes and each of them corresponds to a perfect number.

[Amber G.]

Infosys Prize2024 in Mathematical Sciences is awarded to Prof. Neena Gupta of ISI Kolkata for her work on the Zariski Cancellation Problem, a fundamental problem in algebraic geometry first posed in 1949 by Oscar Zariski, one of the founders of modern algebraic geometry. In 2014 she proved the striking result that Asanuma’s 3-dimensional affine variety yields a negative answer to Zariski’s original Cancellation Problem in positive characteristic.
Congratulations!

[Yogi_G]

Am looking for some review of sri Yagnadevam's work on mathematical proof of IVC script being Sanskrit. The Math theory is beyond my grasp and I am trying to find if it is genuine or snake oil. The fact that his paper is not peer reviewed adds to the angst.

[Amber G.]

Am looking for some review of sri Yagnadevam's work on mathematical proof of IVC script being Sanskrit. The Math theory is beyond my grasp and I am trying to find if it is genuine or snake oil. The fact that his paper is not peer reviewed adds to the angst.

I have put some of my initial thoughts here: viewtopic.php?p=2632710#p2632710

Posting it here for convenience .. may add later some more thoughts:

FWIW Few comments at this point (I may give more insight after reading/ understanding it a little more and see other experts reviewing his work)

Key Highlights

- Yajnadevam employed cryptanalysis methodology to crack the code.
- The detailed manuscript is accessible on Academia.edu.
- Challenges prevailing theories on Aryan migration and language origins.
- Supports ancient chronological estimates for the Vedas and Itihasas.
- Advocated for inclusion in school curricula (Curriculum Significance:).

While some cryptanalysis are meticulously reviewing the manuscript..

IMO his work will face resistance from the Indological establishment as it disproves Aryan migration theories and pushes back Sanskrit classical grammar origins. (Saw M Witzel of Harvard already commenting/criticizing it pathetically - with zero understanding of his methods)

Some "Eminent historians" will likely ignore this evidence but we must study it..

May update with further insights after reviewing Yajnadevam's pioneering work.

[Yogi_G]

Thanks. Do his claims on the mathematical aspects hold good? His claim that when replaced with Tamil the validity doesnt hold Good, is that a valid claim? I saw a few crypto analysis reviews on twitter but more opinions will help the cause. I was super enthused with the content and if true, it demolishes everything we know and have been taught so far.

[Cyrano]

https://www.quantamagazine.org/mathemat ... -20241211/

[Vayutuvan]

https://www.quantamagazine.org/mathemat ... -20241211/

That is very nice. Maybe Goldbach conjecture can be resolved as well using this method. Gower's norm seems to be a nice tool in numerical analysis as well. I am going to study that. Are there people here in the forum who know more about Gower's norm?

[vsunder]

Conjecture 1: I have resolved a 90 year old conjecture in Fluid Dynamics, Aerospace engineering, Mathematical Physics, Physics and Mechanical engineering going back to Lord Rayleigh, Ludwig Prandtl(the father of modern Aerodynamics) and Wilhelm Nusselt the father of Heat and Mass Transfer. Lots of people, physicists and engineers and mathematicians worked on this question doing experiments and numerical simulation for example Leo P. Kadanoff(univ. of Chicago) but finally the rigorous proof is due to me and my friend and collaborator. The article is accepted in one of the great mathematical journals of the world with a 15% acceptance rate.
The referee reports are stunning and absolutely glow.

Conjecture 2: This is a 30 year old conjecture in Geometry that was initiated by two Fields medallists and also got solved by me and this paper has appeared in an outstanding journal which is in the top 5 in the world at the same level as Conjecture 1 journal. One of the Fields medallists one of the outstanding mathematicians of the 20th and 21st century wrote to us with a cryptic congratulatory note.

I shall post details slowly. All the papers are on Arxiv though the one for Conjecture 1 on Arxiv does not incorporate the referee comments.
Conjecture 2 is published.

Regarding Conjecture 1 lots of professors are still doing experimental work in IITs like Umesh Madanan at IIT Kanpur, no need its rigorously proved. Same can be said for Kadanoff and others who took all sorts of materials and did experiments. Game is OVER!!!

[vsunder]

https://www.quantamagazine.org/mathemat ... -20241211/

That is very nice. Maybe Goldbach conjecture can be resolved as well using this method. Gower's norm seems to be a nice tool in numerical analysis as well. I am going to study that. Are there people here in the forum who know more about Gower's norm?

Please do NOT TALK about things you clearly have superficial or zero knowledge about. You want to read serious Number Theory you can read some of my notes online. And it is not my subject yet I have lots of notes. Iwaniec mentioned in the article is my colleague and has his office 3 doors away and is my closest friend and so is Friedlander from Toronto mentioned in the article. Henryk (Iwaniec) will tell me an opinion soon and he can be hard, when we go to lunch next week he will tell me. He is one of the great number theorists in the world.

[Vayutuvan]

Please do NOT TALK about things you clearly have superficial or zero knowledge about. You want to read serious Number Theory you can read some of my notes online.

Will do sir. I am not a mathematician, leave alone an NT chap. That said, I believe that I can grok/understand stuff quite well, thank you. I have my own strengths and you have your own weaknesses and I know what they are. I am 100% sure about it. Anyways, what is this territorial fight? Are there any rules that somebody else cannot do NT other than those who are already in NT? How about a field medalist or other pulling you up, dear professor?

[Vayutuvan]

Conjecture 1: I have resolved a 90 year old conjecture in Fluid Dynamics, Aerospace engineering, Mathematical Physics, Physics and Mechanical engineering going back to Lord Rayleigh, Ludwig Prandtl(the father of modern Aerodynamics) and Wilhelm Nusselt the father of Heat and Mass Transfer. Lots of people, physicists and engineers and mathematicians worked on this question doing experiments and numerical simulation for example Leo P. Kadanoff(univ. of Chicago) but finally the rigorous proof is due to me and my friend and collaborator.
...
Game is OVER!!!

I know all about numerical experiments - computational mathematics. Proving/disproving conjectures through computations is as old as FCT. FCT proof is accepted as valid proof - as valid as what ever your are calling "rigorous".

If you are saying that that computational experiments are not important or computer proofs - large number of cases analyzed using a computer as in FCT - are not rigorous enough, I have to disagree on philosophical grounds. I will give you that they may I repeat may lack elegance, but they are as valid as any "rigorous proof". How elegant is FLT proof, if I may ask.

[vsunder]

Conjecture 2: The paper (the conjecture is due to C. Fefferman (Princeton) and ST Yau(Harvard))

https://drive.google.com/file/d/1Q7G1tz ... drive_link

Letter from ST Yau:

https://drive.google.com/file/d/1Icbii5 ... drive_link

The result is sharp and finishes the conjecture once and for all time. JDG is a really tough journal. Conjecture 1 will appear in a journal in my opinion is even higher than the very high level of JDG. You will see when I post.

https://en.wikipedia.org/wiki/Shing-Tung_Yau

https://en.wikipedia.org/wiki/Charles_Fefferman

This is on the Geometry and Spectrum of the Laplace operator. Conjecture 1 I want to explain in some depth. S. Chandrasekhar the astrophysicist and Nobel Laureate did work on Conjecture 1 too. See his beautiful book " Hydrodynamic and Hydromagnetic Stability" since the phenomenon happens on earth and on stellar atmospheres.

[Vayutuvan]

Since we are on the topic here, let me also say this.

As a reasonably good Computer Scientist who believes in constructivism, noone should be surprised if I take the extreme position that any claimed rigorous proof needs to formalized and verified using at least three diefferent proof assistants.

My favorites are Coq, LEAN, and HOLLight.

Readers should also consider the fact that 100s of thousands of theorems are proved annually. Not all theorems are made equal.

Taking extreme positions in STEM actually is counterproductive. It the spirit of an endeavor which is mildly competitive yet ultimately cooperative in nature. I would be very suspicious of any advisor who starts competing with his/her own students

[vsunder]

Ravi S. Kulkarni is a distinguished mathematician and my friend.

https://en.wikipedia.org/wiki/Ravindra_Shripad_Kulkarni

At a late stage in his career after professorships at Columbia University, Indiana Univeristy and Graduate Center CUNY(City University of New York) he was offered the directorship of the DAE(Department of Atomic Energy) Harishchandra Research Institute (HRI) at Prayagraj. He headed HRI for several years and then became a distinguished professor at IIT Bombay.

https://en.wikipedia.org/wiki/Harish-Ch ... _Institute

DAE had to make an exception for him to assume the directorship since he was US citizen, which happened due to Kakodkar pushing the case After retiring from IIT Bombay he headed the Bhaskaracharya Pratishthan at Pune which was founded by SS Abhyankar a famous algebraic geometer and Marshall professor of Mathematics at Purdue who passed away a few years ago. Abhyankar was one of Oscar Zariski's famous students at Harvard and also a teacher of mine in graduate school at Purdue.

Ravi Kulkarni celebrated his 80th birthday during pandemic times and a conference was held in Pune via Zoom mostly. I spoke at this conference on Conjecture 2 above ^^^ and it was exciting and thrilling to have Yau in the audience. Here is the poster for the conference. Ravi is an erudite scholar of Sanskrit and Indian mathematics and there was also a session on that aspect as you can see from the poster.

https://drive.google.com/file/d/1Cz2RO4 ... sp=sharing

In my young days as a post-doc Ravi was my colleague we both worked late hours and we talked a lot on Hyperbolic geometry, Teichmuller theory and Minimal surfaces which a few years down the road became a powerful tool for me in studies on free boundaries and Composite materials. Those papers on designing Composites are linked in posts on the Mil Forum and have become landmark papers with lots of citations. Conjecture 1 paper will become a landmark paper. Just you watch!!

My lecture on Conjecture 2 was recorded by Bhaskaracharya Pratisthana for the Kulkarni 80th birthday conference and posted on youtube. Here is the link:

https://www.youtube.com/watch?v=nSjH3FpmtTw

Regarding Conjecture 1 I do not want to make several posts, so a few everyday. I will also link my notes on Soap Films and Soap Bubbles that will be a book. I will only link 3 chapters from the book. The notes are 300 pages handwritten and a last chapter has still to be written.

I want to post my correspondences with Padma Bhushan C. S. Seshadri and the repeated attempts of the Tata Institute for me to go to India. Unfortunately it did not happen for multiple reasons. Kulkarni also tried to get me.

https://en.wikipedia.org/wiki/C._S._Seshadri

Both CS Seshadri and MS Narasimhan wanted me very much and had tremendous respect and affection for me. I will have to scan some of the correspondence regarding their offer/s to me and then also why I did not go. This first happened at the age of 29 for me in 1984. I have kept the correspondence carefully in file folders.

You can hear me in the first minutes of my video speak of my young days interacting with Ravi as a post-doc.
Here is the link to Bhaskaracharya Pratisthan in Pune, where the physical conference was held for Ravi.

https://en.wikipedia.org/wiki/Bhaskarac ... atishthana

I am NOT name dropping here as several charlatans and kooks are doing that they know this person and that person. You are all frauds and I will call you out. You are just contemptible zeros who know no Physics, no Mathematics no anything just bullshitters par excellence. You know next to NOTHING about Scientific Education in India, all bullshit and conning people here.

I will also post my papers on Stealth and Invisibility. In the MIL forum so much bullshitt is being propagated by charlatans on Stealth. Those papers were written when I was 32. Every single review article has to quote that paper on Invisibility.

Plus all sorts of notes that the charlatans here will have no leg to stand on.

[Vayutuvan]

@vsundar gaaru (aka Prof. SC), I apologize if I overstepped some sort of a red line. I have always enjoyed your posts and also read your notes on class number field theory. I am a fan of you sir and all mathematicians. Becoming a distinguished professor of math at a university like Rutgers is an achievement I can only dream about.

I Still hold that there pure math is not the end all and the be all. No offense.

[Vayutuvan]

https://www.quantamagazine.org/mathemat ... -20241211/

quantamagazine is somewhat of a hype magazine. It publishes mostly popular math and science. IMHO and all that.

[Vayutuvan]

Please do NOT TALK about things you clearly have superficial or zero knowledge about.

But sire, I can talk about Zero Knowledge Proofs. I also can figure out whether someone does indeed have a proof without knowing the proof itself. In some cases.

[Cyrano]

A teacher without patience is like a cloud that does not rain. May look huge, majestic and takes up lot of space but ultimately is of no use and disappoints.

[vsunder]

Three Numbers in Fluid mechanics and Conjecture 1

There are three important numbers in Fluid Mechanics. The first one is the Prandtl number. This number was discovered by Ludwig Prandtl. When fluid flows over a wing of an aircraft. The layer in contact with the wing of an aircraft say is at rest. At increasing heights the fluid will flow at the same speed of the aircraft. Thus we have a transition layer from zero velocity to the velocity of the stream. This is where Prandtly made his many contributions on the boundary layer. In modern fighters it is important to keep the layer thin or else it will interfere with the flow profile. One does this on the LCA and other aircraft by "bleeding" the boundary layer by making holes in the wing. Bleed air from the boundary layer is channelled to cool the avionics. More on the Prandtl number and Prandtl.

https://en.wikipedia.org/wiki/Prandtl_number

Next comes the Rayleigh number. In thermal convection it involves the temperature gradient of the fluid, that is the temperature difference between the top and bottom and also involves the viscosity and some dimensions of the container holding the fluid. Here is more about the Rayleigh number.

https://en.wikipedia.org/wiki/Rayleigh_number


Lastly comes the Nusselt number from Heat and Mass Transfer

https://en.wikipedia.org/wiki/Nusselt_number


Here is something about Lord Rayleigh, Ludwig Prandtl and Wilhelm Nusselt.

https://en.wikipedia.org/wiki/John_Will ... n_Rayleigh

https://en.wikipedia.org/wiki/Wilhelm_Nusselt

https://en.wikipedia.org/wiki/Ludwig_Prandtl


For over 90 years a scaling relation was suspected between these three numbers whenever connection takes place. A model for convection is the one proposed by Lord Rayleigh and Benard. It goes by the name of Rayleigh-Benard convection. One can see the convection rolls in clouds as in these pictures, these are Rayleigh-Benard convection rolls or the hexagonal cells in boiling soup as in these videos.

https://www.youtube.com/watch?v=nQUH9nGTZTY

https://www.youtube.com/watch?v=gSTNxS96fRg

https://earthobservatory.nasa.gov/image ... he-pacific

These conjectures were made by Lord Rayleigh, butone finds them in the works of Markus and LN Howard of MIT who were people who worked in Fluid Mechanics. Their papers date from the mid 1950s. Another recent reference is from the paper of Leo P. Kadanoff where the conjecture is stated explicitly( see eqn. 3(b))

https://pubs.aip.org/physicstoday/artic ... ctures-and

So what is the conjecture. Here is the statement and what has been proved: At high Prandtl numbers



Nusselt number < 0.3( Rayleigh number )^{1/3}.


There is also a corresponding statement for low Prandtl numbers which is a manuscript which is being written up.

Here are some slides of a lecture given in June. They give a nice description of the central ideas and the techniques that went into the proof of the conjecture.

https://drive.google.com/file/d/1V8ndrW ... drive_link

Here is some additional comments by the reviewers of the paper for the Inventiones Mathematicae. In particular they note that rigorous investigation of the inequality started some 20 years ago, and they have also given some more history of the problem which one also finds in the introduction of the our paper:

https://drive.google.com/file/d/1nA3q98 ... sp=sharing

https://drive.google.com/file/d/1v4mZEi ... drive_link

[vsunder]

I have scanned ,my correspondence with CS Seshadri a Padma Bhushan and the preminent Indian mathematician along with MS Narasimhan both of whom I knew very well. Seshadri tried on multiple occasions to move me to India and so too did Narasimhan and Kulkarni in leadership roles at leading DAE(Department of Atomic Energy) institutes in India which are the top echelon in India way above the IITs which are community colleges for coolie education. Such appointments at the DAE are to be vetted by the PM's office, as the DAE is a portfolio always under the PM of India. These attempts lasted till 2010 but for multiple reasons I did not go. I have also scanned letters from the Director of the TIFR and I am just thinking of putting them in public. Most likely I will release them. Seshadri and Narasimhan are no more.

Here I wish to talk about my work on White Dwarf stars and post all that work and my reminiscences of interacting with the Nobel laureate S. Chandrasekhar when the first of my two papers were written. He liked the paper immensely and the paper appeared in the top journal for Mathematical Physics as you can see. The two papers are also listed as [64], [65] references in the wikipedia entry for White Dwarfs.


https://en.wikipedia.org/wiki/White_dwarf

Chandra pointedly observed that we were very careful in assigning credit in the intro. A third paper on Rotating Neutron Stars also called Pulsars never got written eventhough all the calculations are done. The culprit is the pandemic and it will get written. Here the situation is more complicated as the treatment is fully relativistic involving the full Einstein equations.

The first paper combines two of Chandra's interests which resulted in two important books. Introduction to Stellar Structures(1938) and Ellipsoidal Figures of Equilibrium(1965) that stems from the Silliman lectures he gave at Yale and is a historical account of the subject. As he told me, "your paper is the first rigorous proof of the compressible case, my papers and the book deal with the incompressible case which is NOT the true situation for stars. It is like the play Hamlet, the Prince of Denmark being performed without the Prince appearing. You have the equation of state and so it is truly the situation inside a white dwarf".
He always had the manner of speaking like a Cambridge don. Unfortunately he was not alive when the second paper appeared, but he was very well aware of the difficulties and I had broached the fact I was embarking on the study of free boundaries for stars with him and he encouraged me. Here are the two papers:

https://drive.google.com/file/d/1oKcyMD ... drive_link

https://drive.google.com/file/d/1qNd4yQ ... drive_link

[vsunder]

Again I want to state that ALL the people who post on Physics thread, Education thread and the posters on Mathematics are all charlatans and Internet personalities, throwing names that they know X, Y and Z when in fact they are zeros.

You want to know the India story, the French story the Israeli story etc etc. here is the India story. It is what I saw, if some your "idols" will appear to have feet of clay so be it but this is what I saw. Enjoy all the correspondence below. I had to think a bit before I released it but I have to so that all these charlatans will lose credibility and stand exposed as crooks.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

First of all to understand things, here is my citation when I was elected a Fellow of the American Mathematical Society(AMS) a decade ago almost. It cites a number of my papers and some of them were written at the age of 21 and 23 etc. The citation mentions these were written as a student and so does the acknowledgement in the paper at the bottom. I will provide links. I had no supervisor at Purdue, the papers were all thought out and solved by me. Looking back I feel I should have written the introduction better and they betray a young author with no guidance from anyone. By now the cite count of the papers has increased a lot. More about that below. Here is the citation from the AMS:

https://drive.google.com/file/d/1SyysVh ... drive_link

Here is the actual award: January 2017

https://drive.google.com/file/d/1EhFF9A ... drive_link

I knew that I had proved a very importantand influential inequality already and so it is no wonder that this paper has 539 citations on Google scholar a single author paper written at 21:

https://drive.google.com/file/d/1O1aJJj ... drive_link

Here is the google scholar link and you can see that the 330 citations in my citation from AMS above has become 539 now.

https://scholar.google.com/citations?us ... AAAJ&hl=en

There was yet another paper I wrote as a student with zero help from anybody. Again the introduction betrays the author is young. I suppose I was more excited I would a way to dice and decompose functions in ways to adapt it to the Hilbert transform. In a few years these ideas would prove extremely potent in resolving a deep and hard conjecture. More about it in a later post. Not today.

https://drive.google.com/file/d/1jpXzho ... drive_link

At some point I wrote this little piece about the article above to motivate my students to attend seminars and here it is:

https://drive.google.com/file/d/1GPqa-k ... drive_link

This paper written at 23 also is referenced in the wikipedia link for Harry Pollard:

https://en.wikipedia.org/wiki/Harry_Pol ... ematician)


https://adsabs.harvard.edu/full/1985CeMec..37..349S

The papers above had already brought me to the notice of many people as someone who was completely independent at such a young age and in their words could produce what they thought was difficult work. So it is not surprising that soon after getting my PhD I was given membership to spend a year(all positions are a year long and are called memberships) all paid at the famed Institute for Advanced Study(IAS) at Princeton, Einstein's place. This is separate from Princeton University across town and there is a gentleman's agreement not to poach faculty from each other. My office was on the ground floor of famed Fuld Hall and next to Einstein's old office with its three large bay windows and only a wall separated me from Einstein's office now occupied by the mathematician Robert Langlands. A long corridor in front of my office ended at the director's office now Harry Woolf and in old days the office of J. Robert Oppenheimer. Nothing is expected of you at the IAS, you need not talk to anyone go to a seminar or even write a single paper. At the end you write a report to the director as to what you did. I must say Ohio State University at Columbus, Ohio had promised me a tenured position and I had accepted and the 1 year at IAS was allowed by them as it is a very prestigious appointment. Here is the IAS apointment letter and also a letter from the director of the Tata Institute from 2004 offering me a 3 year adjunct position. This is a DAE position and all that:

https://drive.google.com/file/d/1lBugB9 ... drive_link

So far so good. I would go to the IAS for one year and then go to Ohio State after that. Ohio is a real "sh...t hole" and Southern Ohio even a worse "sh...t hole". Some of you claim you work there, and know big people. Think about it, these big people could not remove you from there, you must be really dumb and stupid. The only thing in Southern Ohio is bootlegging along the Ohio river in illicit stills.

But then something happened. There was this very difficult problem in Harmonic Analysis and people at Princeton University were working on it. I started thinking about it at IAS. I realized I could use clever decompositions of functions and couple that with Algebraic Geometry that I had learned as a student from Abhyankar at Purude, to understand these phase functions that appear in the problem. In one variable such phase functions appear in the work of Hardy-Littlewood-Ramanujan in their famous circle method. So bang the problem got solved and the paper appeared in one of the top journals of the world. Even my AMS citation mentions that work and states itis now in textbooks and monographs. Here is the paper and you can google and check that Duke J. is a great journal:

https://drive.google.com/file/d/1Zyq4xx ... drive_link

So at the end of my year at IAS, I started to get feelers to move. However I did go to Ohio state but it was obvious they my days there were numbered and several universities were wanting to get me. This is 1985 Fall. The first one to want me was Rutgers a powerhouse in Mathematics and UCLA, Univ. of Notre Dame and Univ. of Illinois at Urbana-Champaign. Into this came a cable/telegram from CS Seshadri that he too was very interested in moving me to the new DAE institute he had taken over at Chennai. This was the old Matscience run as a family enterprise. The DAE had taken it over and appointed EC George Sudershan as the director with CS Seshadri as the co-director. Given that I had tremendous respect for both Seshadri and Narasimhan I could not say no out of hand and so decided to fly to India and take a look. Here is the Seshadri correspondence

https://drive.google.com/file/d/1jOzAXw ... drive_link

More about Padma Bhushan CS Seshasdri

https://en.wikipedia.org/wiki/C._S._Seshadri

Arriving in Madras on my own money, I had to fend myself finding a hotel and conveyance to go everyday to the Institute. Venu Srinivasan the TVS man helped me. We are close friends. He did his MBA from the Krannert school of management at Purdue when I was there. He stayed in university housing and was a vegetarian and use my kitechen to cook sambar every weekend and drive me crazy using up every pot and then I had to wash it. He also smoked Sobranie black Russian cigarettes and made me smoke more and his daily routine would call for arriving at 1 am in my office to round up whoever was working late in the building for touch football and soccer in the Mall outside. He offered to put me up in his lavish, palatial mansion 2 AdyarClub Gate Rd. which I refused and so he got me discounted rates at Taj Connemara and a driver and car to go to the Institute. I remember the first time he came to see me at Woodlands where I was statying, the whole staff started scurrying around and we went to the restaurant and the manager waved his hands and people just ran away and there was no one within 50 meters from us.
Now he is a Padma Bhushan too and I wrote to him congratulating him when Modi govt. announced this and reminded Venu of his oil guzzling aluminum engine Vega he drove at Purdue which I had christened the Oilmonger.

Anyhow the first day I walked into IMSc I met Kumaresan a close friend and familiar face who calls me "The Boss". Kumaresan has done yeoman service now as the Director of MTTS, the Mathematics talent and Training program. I knew Kumaresan 1981-1982 when he spent a year at Indiana University and I enjoyed discussing Representation theory and Lie groups with him that helped me in future work on Symmetric spaces( also mentioned in my AMS fellow citation). Kumaresan is a TIFR product who left TIFR and was faculty at Mumbai University. Later towards the end of his career he joined the Central University of Hyderabad and now he is retired and lives in Vizag from where he runs the MTTS.


https://mtts.org.in/s-kumaresan

Kumaresan warned me that Seshadri and George Sudershan were at loggerheads. Sudershan would stay 8 months in Austin, Texas, while the Institute work would languish as Sudershan was he director and then in the winter Sudershan would come to India. Everytime, Sudershan left town and came back, the office peon would garland him. Sudershan was in residence as it was January when I visited and Kumaresan promised me that the next day I would see this garlanding ceremony as Sudershan was due back from Mumbai. True as enough as soon as Sudershan stepped out of his official Ambassador, the peon rushed to garland him with marigolds. It was a nauseating spectacle. I met Seshadri and had a pleasant conversation with him. He was very eager to make an appointment and said housing was not available at the Institute but was easy in Chennai etc. You can read his letters to me where he reiterates that. But my meeting with Sudershan and ex officio IAS officer was all one sided as I said nothing as my mind was made up. Soon after in 1989 Seshadri resigned from DAE and IMSc and joined CMI which was started by SPIC group of MA Chidambaram, so in hindsight I am glad I did not accept.

https://www.cmi.ac.in/seshadri/tributes.php

IMSC is now a useless place. All rot.

https://www.imsc.res.in/

In Seshadri's letter to me he writes he asked F. Treves from Rutgers an opinion about me. Treves tells him that Rutgers is also interested in me and starting proceedings to get me. Seshadri's letter is addressed to me at Ohio State. In the end I moved to Rutgers and Treves became my colleague and my co-author again a landmark paper in one of the best journals in the world which also finds mention in my AMS citation.
Treves and Seshadri knew each other well when Seshadri was in Paris studying wih the famous French algebraist Claude Chevalley and Treves with the Fields medallist Laurent Schwartz in the 1950s. MS Narasimhan also studied with Laurent Schwartz.

https://en.wikipedia.org/wiki/Fran%C3%A7ois_Tr%C3%A8ves

https://sites.math.rutgers.edu/~chanillo/solv.pdf

India did not stop. Another attempt was made as you can see in the TIFR director's offer to me in 2004. The idea was that I take 3 years to see the situation by making visits every year and then decide. But by then I had deep roots in the US and it was not possible though I spent considerable time in TIFR (Tata Institute for Fundamental Research)and continue to help them. This is also stated in my AMS citation.

For example I ran a 3 week summer school followed by a 15 day workshop 6 months later for the DAE supported NBHM(National Board for Higher Mathematics), the TIFR and IIT Bombay which was a major effort as I wrote the syllabus and got the organization done exccept for budget and local arrangements. I whave since helped NBHM 5 times. Here is the poster for the first time I did this in 2013.

https://drive.google.com/file/d/10rDvys ... drive_link

By the way The Institute for Advanced Study(Princeton) has invited me two more times after the first visit 1984-1985

https://www.ias.edu/scholars/sagun-chanillo

These were special years on Conformal and CR Geometry etc. They even did a Press release when I was elected an AMS fellow here it is

https://www.ias.edu/news/ias-scholars-n ... al-society

[Vayutuvan]

Prof. @vsunder,

So at the end of my year at IAS, I started to get feelers to move. However I did go to Ohio state but it was obvious they my days there were numbered and several universities were wanting to get me.

What do you think of Reverse Mathematics program started by Friedman? He is a logician, different field from yours. As a TCS man, I find it very intriguing.

:praNaam:

[Vayutuvan]

@Prof. @vsunder, one more question. Why did you turn down UIUC? IIRC, it had a great faculty at that time and was also quite high up in the ranking but fell from those heights in later years.

[vsunder]

The resolution of the conjecture treated in the linked paper took us 10 years to prove, persistence and doggedness. The proof of the conjecture takes close to 80 pages as you can see. It is a basic and classical question we resolved. Take a function on the interval [-1,1]. Expand it as you do Fourier series but instead use another orthogonal system, the Jacobi polynomials. Legendre polynomials and spherical harmonics of fundamental importance in Engineering, Physics, Chemistry etc are special cases. Then when does the Legendre series or Jacobi series etc converge to the function? Does it? when does it fail. Astoundingly we got a precise answer after a lot of work. The referee report that is attached is also glowing and blunt. It did not take us long with the phrasing to guess who the referee was. It was the late great Richard "Dick" Askey a world authority on hypergeometric series and Special functions. I began my interest in Legendre Polynomials with Harry Pollard about whom I have said earlier. This paper which solved a major conjecture was the last I worked in this field. I always try to find the hardest problem in a field within my capabilities. Solve it and leave the field forever and do something radically different.

Here is the wikpedia page of Askey the referee whose report I have linked below:

https://en.wikipedia.org/wiki/Richard_Askey


Askey's referee report:

https://drive.google.com/file/d/1noZtPv ... drive_link

Here is the long, long, long proof of the theorem. I have saved you $50.00 too. as that is what the American Mathemaical Society will charge you for a Memoirs off print.

https://drive.google.com/file/d/1jFwSsq ... drive_link

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The theorem proved above has become very important in the study of the Fekete problem which can be also considered to be a problem in Electrostatics. Stephen Smale a famous Topologist, who solved the Poincare conjecture in 5 and higher dimensions and was awarded a Fields medal for this work, has really pushed the Fekete problem. There are couple of papers recently posted on Arxiv using our work towards solving the Fekete problem. But I am NOT going back to thinking about this area anymore,

https://en.wikipedia.org/wiki/Fekete_problem

https://en.wikipedia.org/wiki/Stephen_Smale

Poincare conjecture in 4 dimensions was solved by Friedman who also got a Fields medal and the dimension 3 case which was the case Poincare asked by Perelman who also got a Fields medal.

[sudarshan]

This equation would be true for any year N for any angle theta:

(cos(theta) + i * sin(theta))^N = cos(N * theta) + i * sin(N * theta)

However, this equation would only be true once every four years, and the year 2025 is one of them:

(sin(theta) + i * cos(theta))^N = sin(N * theta) + i * cos(N * theta)

So for any value of theta:

(cos(theta) + i * sin(theta))^2025 = cos(2025 * theta) + i * sin(2025 * theta)

and

(sin(theta) + i * cos(theta))^2025 = sin(2025 * theta) + i * cos(2025 * theta)

[Amber G.]

^^^ For

(sin(theta) + i * cos(theta))^N = sin(N * theta) + i * cos(N * theta)

:
True for any n of the type 4k+1 .. (ratio goes as 1, i , -1 , -i ..)
BTW modulus of both sides is still the same for all the values of integer n

For 2025
2025 = (0+1+2+3+4+5+6+7+8+9)²
= 0³+1³+2³+3³+4³+5³+6³+7³+8³+9³

Happy New Year!

Okay, just an easy one: Arrange them in order: (Which one is the greatest?): (All quantities are absolute values)

[sudarshan]

Okay, just an easy one: Arrange them in order: (Which one is the greatest?): (All quantities are absolute values)

Do they all have an absolute value of 1?

[Amber G.]

^^^ ..
Yes! This was meant to be a simple yet sort of amusing clever problem. Here's the twist:
e and π can be any real numbers!
The magic happens when you raise:
i to any real power (i^(any real number))
or
any real number to an imaginary power ((any real number)^(i*something))
The result of absolute values is always the same!

[Amber G.]

Happy New Year

For a fun challenge, here is an ancient problem from the land of mathematical giants Brahmagupta and Ramanujan!

313 is considered an auspicious number. It's a "Happy Number", a palindromic , a prime, and an "emirp" (a prime number whose reverse is also prime). Not to mention that 313 = 12²+13².

Here's your challenge:


Find two natural numbers (positive integers) a and b such that:

a³ + b³ = 313² × c³

where c is also a natural number.

Rules:
Feel free to use any resources at your disposal, including:

- Internet research
- Calculators
- Computer programs
- Old solutions of similar problems here in this math dhaga (:)
- AI tools like ChatGPT , Meta etc (see if they can successful)
- Consult your favorite Math Professor

I won't provide a solution for some time, so take your time to think and experiment.
If you've seen this problem before (or a similar one), just give your answer (and a brief comment) but no link or details of the solution.

Wait for a few weeks before sharing detailed solutions.

Let's see how powerful the current state of AI and computer programs can be when combined with the collective brainpower of BRF!

[Vayutuvan]

Grok:

My prompt: What are a, b, and c whole numbers where a^3 + b^3 = (c^3) * 313^2?

Grok Answer:

To find whole numbers a, b, and c such that a^3 + b^3 = (c^3) * 313^2
, let's break this down:

1. First, calculate 313^2: 313^2 = 97969


Let's look for simple cases where a, b, and c are could small whole numbers.

1. If c =1, a^3 + b^3 = 97969
There are no simple whole numbers a and b (... which satisfy).
because 97969 is not close to any sum of cubes of small numbers.
If c = 3, a^3 + b^3 = 2645163
This number is too large for simple calculations without computational help, but we can try some modular arithmetic or look for patterns in smaller numbers.
Let's try another approach with modular arithmetic to see if we can find a pattern. We'll check modulo 9 since
313 ≡ 1 (mod 9), 313^2 == 1^2 == 1 (mod 9)
Here are the cubes mod 9: ...
Therefore, we need:
a ^ 3 + b^3 = c^3 (mod 9)
Let us test for small cubes mod 9. ...�
..., so this doesn't work.
A systematic approach would require checking numerous combinations or using computational tools. However:
One known solution is: a=426, b=465, c=321.
...
This problem requires a significant amount of computational power or a deep dive into number theory for more solutions. Therefore, one known set of
...
Finding other solutions would involve further mathematical exploration or computational searching.

[Vayutuvan]

Grok gave the following code

Code: Select all

 
from math import isqrt

def is_perfect_cube(n):
    cube_root = round(n**(1/3))
    return cube_root**3 == n

def find_solutions(max_value):
    for a in range(1, max_value):
        for b in range(1, max_value):
            sum_cubes = a**3 + b**3
            if sum_cubes % 97969 == 0:  # 313^2
                c = isqrt(sum_cubes // 97969)
                if c**3 == sum_cubes // 97969:
                    print(f"Solution found: a={a}, b={b}, c={c}")

# Example usage
find_solutions(1000)  # Adjust the upper limit as needed

[Vayutuvan]

Another solution

(927,496,616)
...
Finding all solutions would require an exhaustive search or a deep dive into number theory, possibly involving elliptic curves or other advanced algebraic techniques. If you have access to computational resources or know of specific algorithms for such problems, you could explore further.

[Vayutuvan]

Grok spit out some more examples when I asked it to give more examples. It also spit out another Py script. Here it is:

Code: Select all

from math import isqrt

def is_perfect_cube(n):
    cube_root = round(n**(1/3))
    return cube_root**3 == n

def find_solutions(max_value):
    for a in range(1, max_value):
        for b in range(a, max_value):  # Start from a to avoid duplicates
            sum_cubes = a**3 + b**3
            if sum_cubes % 97969 == 0:  # 313^2
                c = isqrt(sum_cubes // 97969)
                if c**3 == sum_cubes // 97969:
                    print(f"Solution found: a={a}, b={b}, c={c}")

# Example usage
find_solutions(1500)  # Increase this for broader search
[code]

[quote]
Advanced Techniques: Using elliptic curves or other algebraic structures in number theory could potentially yield more solutions or patterns, though this would require deep mathematical insight.

Please note, these examples are not exhaustive. Each search might reveal new solutions, and the complexity grows with larger numbers. If you need more solutions, you might need to scale up the computational effort or dive deeper into number theory.
[/quote]

[Vayutuvan]

It is not that interesting to me beyond this point. It is some diophantine equation (DE) solution of which is not all that important. What is important is the thought process of Grok. It is pretty decent.

I don't think it is a good use of somebody's time. Time would be better spent in pursuing an NT two-course sequence at OCW or some other online platform to be able to write algorithms for DEs of certain forms.

[Vayutuvan]

MRDP Theorem shows that there is no general method to solve named for (Yuri) Matiyasevich-(Julia) Robinson-(Martin) Davis-(Hillary) Putnam shows that there is no general algorithm, i.e. it is undecidable, to decide whether a given Diophantine equation has a solution (leave alone finding that solution even if a solution exists). This was Hilbert's 10th problem which has been resolved in the negative by Matiyasevich after decades of progress made by RDP.

The following post is a lot more interesting for computer scientists and Computational Mathematics (which is different from Computational Science /Engineering or what have you - Computational Chemistry/Physics/Quantum Chemistry/Biology/Genomics/Proteomics/etc.)

https://math.stackexchange.com/question ... -equations

Take a given Diophantine equation. Chances are, we can't find any solutions. But if it's an equation of a certain form, we may get lucky and may be able not only to find a solution, but be able to classify all the solutions of such an equation.

I'm preparing a talk on Diophantine Equations. I'm interested in exploring how the difficulty of Diophantine equations increase as both the number of variables and the degree of the equation increases. Of course, due to Matiyasevich's resolution of Hilbert's Tenth Problem, we know that no general algorithm to determine whether or not a given Diophantine Equation has a solution.

Here's my current understanding of the "frontier" of Diophantine Equations:
---------------------------
Variables =1, Degree =n
The solutions are completely determined by the Rational Root Theorem. Once you have all the rational roots, just look for the roots that are also integers.
---------------------------
Variables =k, Degree =1:
The equation is of the form
a_1 x_1+...+a_k x_k = d.
This has solutions if and only if the greatest common divisor of a1,...,ak divides d, and in that case, we have a nice formula parametrising the solutions (which I won't repeat here).

---------------------------
Variables =2, Degree =2
There exist solutions to an equation P(x,y)
if and only if we have:
1) There exist solutions to P(x,y) modn for all integers n (so in practise, you just need to verify there exists solutions mod p for all primes p, and then by Hensel's Lemma, and the Chinese Remainder Theorem, we have solutions for all n

2) There exist real solutions to P(x,y) If we can find one solution, we can find all rational solutions through stereographic projection. Once you have all rational solutions, look at the solutions where the denominator divides the numerator to get all the integer solutions.

---------------------------
Variables =2, Degree =3

This is handled by elliptic curves. Broadly speaking, provided that the equation is "nice enough", the set of solutions to the Diophantine equation has an abelian group structure. This consists of a torsion free part, which can be one of 16 possible groups, and a torsion free part, which looks like \matbb[Z]^ for some natural number r. The r here is called the rank of an elliptic curve, and computing it is, in general, computationally difficult.

---------------------------
Variables ≥2, Degree ≥4:

In general, this is no man's land. Of course, some equations we may have luck with (take x4+y4=z4; there is an elementary proof that this has no solutions in the natural numbers), but in general, you're probably out of luck.

The other two types of Diophantine equations that I haven't spoken about are exponential Diophantine equations (such as Catalan's conjecture: xa−yb=1 only has one solutions, namely x=3,a=2,y=2,b=3) and infinite Diophantine equations. These are even more intractable, so I won't touch on these at all.
...

[Vayutuvan]

The equation given by @Amber G ji has 3 variables (a, b, and C) and degree 3. If it can be reduced to 2 variable degree 3, (How?), then maybe one can use the method given above.

As far as I am concerned, this particular problem is not something that would hold my interest.

[Vayutuvan]

https://math.stackexchange.com/question ... -equations

This question and its answer are somewhat interesting.

There's no algorithm that correctly decides if a Diophantine equation does or doesn't have a solution. Still, many equations can be successfully analyzed, and I'm wondering if anyone wrote down a "cookbook" for dealing with Diophantine equations of various shapes and forms, including the higher-degree, higher-dimensionality ones.

Given a system of polynomial equations with integer coefficients, we may wish to determine if there are any solutions in integers, and if so, whether there are finitely or infinitely many, and whether they can be explicitly described; we may also wish to determine if there are any solutions in rational numbers, and if so, whether there are finitely many, etc.

If the system is linear, do this (easy).
If there is just one variable, do that (easy).
If there's one quadratic equation in two variables (or a homogenous one in three variables), there's again an explicit procedure: check if there's a singularity, determine if there are integer solutions at all (Hasse Minkowski), parametrize the curve, etc. I think all questions can be effectively answered in the case of genus 0 curves.
If it's an elliptic curve, follow these steps... (I don't think all questions can be algorithmically answered, at present).
Higher genus curve? What do you do? Find the Jacobian? What else?
Higher dimensional surfaces and varieties? What do you do? Which heuristics do you try, what are some useful families of equations that can be attacked?
All of those pieces are well covered in the literature - I'm just wondering if there's a good resource that succinctly describes the various alternatives that we may be able to handle.

Note: this older question has similar goals, but it stops short of giving details on how to handle genus 0 and genus 1 and says nothing much about higher genera and higher dimensional varieties.

...