BR Maths Corner-1

[Amber G.]

Vayutuvan - Thanks for all the posts.. few comments..
Some of us have tried popular AI tools for this math problem (and similar ones), but the results often came out bizarre - yielding explanations/solutions that are laughably absurd, camouflaged in technical-sounding language that ultimately translates to nonsensical gibberish.

However, the responses from Grok by Vautuvan ji take it a step above - using nonsensical 'technical mathematical' language to spit out easy-to-check wrong answers, thus demonstrating a staggering lack of mathematical understanding.

For example - A glaring example is the claim that "one solution is a=426, b=465, c=321...

Grok:

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


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 ...
[Skipping lot of pointless parts using impressive but meaningless/unhelpful language]

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.

.... easy to check using basic math that it is wrong!!! . Yet, the response boasts of a "systematic approach" despite being utterly wrong!

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.

This pattern continues with subsequent "solutions" that are equally flawed, (and easy to check with basic math) accompanied by outlandish claims of mathematical expertise.

-/sigh/ -

Then it also mentions a "computational solution" with programming code (a novice programmer can produce such a 'code' , begging the question: why not utilize this code to derive at least one correct solution?


With all that .. it presents NO solutions (which is easy to verify) with, IMO zero mathematical insight.
So, when all said and done, is there a solution - which can be easily checked with ordinary math?

[Vayutuvan]

I haven't checked the correctness of the solutions. I thought that Grok at least backcalculated. Does that Python code work? I have no idea. I haven't gone through the code. LLMs are not intelligent. I doubt they can get mathematical insights a mathematician working in that area can get.

Some problems are simple to state in day-to-day to English assuming some amount of general knowledge. But the solutions can be very hard requiring carefully defining the all the terms, notations, and axioms.

[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]

For example - A glaring example is the claim that "one solution is a=426, b=465, c=321...
...

Maybe Grok checker (if there is one at all) doesn't implement large number arithmetic. But checking this particular answer for correctness needs at most 10 digits which is definitely doable with signed 64 bit integers. It compute upto 19 digits.

There is some discussion in the mailing list FOM - Foundations of Mathematics - of crafting chess problems after feeding one or more of the LLMs with some chess database. I will post those links since there are many here who are also interested in Chess itself.

[Vayutuvan]

One last comment on that Grok answer to my prompt:

At least Grok was able to "recognize" this as a problem in Number Theory and it correctly "recognized" the equation to be satisfied. But then the recognition was pseudo-recognition in the sense that if it really did recognize it as an equation to be satisfied, whatever solutions it gave it should have cross-checked that they satisfy the equation.

So it really did not recognize (or learn) the problem posed as we human do.

This whole LLM stuff makes me think that the next AI winter is going to last for a long time.

[Vayutuvan]

https://arxiv.org/pdf/2002.00888 by Bruce Reznic

In 1913, Ramanujan posed to the Journal of the Indian Mathematical Society the following question:
“Shew that (6x^2 − 4xy + 4y^2)^3 = (3x^2 + 5xy − 5y^2 )^3 + (4x^2 − 4xy + 6y^2 )^3 + (5x^2 − 5xy − 3y^2)^3"

That is easy to do obviously. What are the generalizations, i.e. replace the numerical coefficients with parameters.

[Amber G.]

^^^ Some time ago, you mentioned Prof Bruce Berendt in this thread. As I indicated then, the problem I asked would likely bring a smile to his face, as it would be very easy for him to provide a solution. It will be interesting to see his reaction.

In general, many problems related to the topic/article mentioned by the article you posted above are still open questions in number theory (NT). For example, while we know/prove that any integer can be represented as a sum of three cubes of rational numbers, we don't know if this is possible using only integers. In fact, if n is congruent to 4 or 5 modulo 9, there is no solution, but for other cases, the solution remains unknown.

The problem I asked is a specific case using sum of cubes and number 313^2, and it's for fun to see how much further we can go using computers, AI, and mathematical discussions in an open forum. We'll see what happens.

[Amber G.]

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.

Vautuvanji - Thanks. Few some frank/honest comments:

- I agree that it's perfectly fine to say 'I'm not interested' in a particular topic, especially when it comes to complex math (and outside to one's interest) problems. However, I find it surprising that you've invested time in "analyzing" the "solutions" in posts after posts, despite acknowledging its lack of importance.

What's even more puzzling is that this "analysis" using Grok contains simple math errors, demonstrably wrong solutions, and absurd conclusions. It's odd to see such a detailed critique, especially when it doesn't contribute meaningfully to the discussion.

Regarding the 'thought process of Grok,' I disagree with your assessment. The analysis contains basic math mistakes and meanders into nonsensical technical jargon. It's hard to see how this thought process can be considered 'decent' when it's plagued by errors and absurdity. (Some other AI's do a fairly decent work)

Yes someone is interested in exploring "Diophantine equations", I'd too recommend following your suggestion to pursue a number theory course sequence on OCW or other online platforms. That would provide a more structured and accurate learning experience.

But I believe that engaging with fun math problems like these can spark a lifelong love for some in mathematics and sometimes lead to serious educational pursuits. Personally, I was fortunate to have brilliant math mentors when I was growing up who inspired me with such problems. I loved such topics even before I took my first number theory graduate level course at IIT Kanpur in mid 60's taught by Prof. JN Kapoor in Math dept. (IIT then allowed us to take courses in other departments). As a math mentor myself, I've used similar problems to foster a love for math in bright youngf students who enjoy the challenge.

Again thanks for that nice paper by Bruce Reznic (written as a birthday gift for our friend Prof Bruce Berendt ). The problem I posted has *very* much relevance to that subject.. but more of this later.

[Vayutuvan]

Vautuvanji - Thanks. Few some frank/honest comments:

- I agree that it's perfectly fine to say 'I'm not interested' in a particular topic, especially when it comes to complex math (and outside to one's interest) problems. However, I find it surprising that you've invested time in "analyzing" the "solutions" in posts after posts, despite acknowledging its lack of importance.
...
Again thanks for that nice paper by Bruce Reznic (written as a birthday gift for our friend Prof Bruce Berendt ). The problem I posted has *very* much relevance to that subject.. but more of this later.

Just to clear up where I am coming from.

I am interested in the ChatGPT itself rather how it solves a problem. Evaluation of the solution to the problem is done by a domain expert. In this case, a number theorist. My views are similar on Chess as well. I can play chess, I know all the rules. But I am not a good chess player as I never spent any time on that aspect. But I am interested in chess playing programs and the algorithms that go into chess playing as well programs that play other games. IOW, my interest is in abstracting the general strategy and algorithmics building clocks of software that plays a game whose rules are formally specified to that program.

A program for playing Chess is a specialization of that general program. You make it faster and better by various modifications such as feeding the program databases of opening and closings. That is where the high ELO rated chess players come in.

Same in this instance too. I am not interested enough to spend a lot of time on NT to be able to be able to evaluate performance of Grok (or any other LLM) on this particular problem. The trick is to come up with a set of problems in this domain to see how well these LLMs do on that ensemble. The ensemble should include a variety of problems from that domain. A domain expert would know how hard or easy a problem of is while going in.

[Amber G.]

<snip>

Just to clear up where I am coming from.

I am interested in the ChatGPT itself rather how it solves a problem. Evaluation of the solution to the problem is done by a domain expert. In this case, a number theorist. My views are similar on Chess as well. I can play chess, I know all the rules. But I am not a good chess player as I never spent any time on that aspect. But I am interested in chess playing programs and the algorithms that go into chess playing as well programs that play other games. IOW, my interest is in abstracting the general strategy and algorithmics building clocks of software that plays a game whose rules are formally specified to that program.

A program for playing Chess is a specialization of that general program. You make it faster and better by various modifications such as feeding the program databases of opening and closings. That is where the high ELO rated chess players come in.
<snip>

Just to clarify, my point is this: Imagine analyzing AI's chess gameplay. One can discuss opening databases, strategies, algorithms, ELO ratings, and more. However, if the AI makes basic mistakes, like incorrect Knight moves, I'm not impressed.

What I found odd about the math problem I provided is that the AI struggled with a simple arithmetic process, requiring only elementary school skills, and erred so confidently. Despite the AI's technical jargon, its mistakes undermine its credibility.

Anyway my challenge is not analyze any particular AI, but to see if problem I posted – involving just simple math could be solved here by collective resource by readers here
As said, one can use, calculators, computer programs, AI, internet search or even asking your math experts. Want to see how long does it take.

[Vayutuvan]

Anyway my challenge is not analyze any particular AI, Want to see how long does it take.

That my dear friend is the problem. Once you make it a challenge it stops being fun. Maybe you like being overly comptetive but not all folks like the same.

Speaking for myself, I want to be competitive enough to make life interesting but not too much competitive that it starts looking like a chore.

[Amber G.]

That my dear friend is the problem. Once you make it a challenge it stops being fun. ....
.. I want to be competitive enough to make life interesting but not too much competitive that ...

...it starts looking like a chore.

LOL!
.. No worries, my friend! No one is suggesting a compulsion here. If you're interested and feel like taking on the 'challenge', great! If not, no problem at all. Just a casual invite for those who'd like to give it a shot (or not).

Here is the problem (again put below for convenience) :
.

Here's your challenge (for fun):


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

[vsunder]

Here is the published version of the paper in Inventiones Mathematicae on the resolution of the Nusselt number conjecture at high Prandtl values. Springer Verlag has made a tweet about the paper on X and also on Facebook. Here:


https://www.facebook.com/SpringerMath/p ... 524594199/

https://x.com/SpringerMath/status/1891356942532002026

About the journal

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

The paper (also can be accessed via the tweet link displayed above also which takes you to open access download from the publisher's web site):

http://sites.math.rutgers.edu/~chanillo ... evised.pdf

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

Also here is a new paper which is for my friend and collaborator Bernard Helffer for a festschrift . Helffer has been the president of the Societe Mathematique de France which is the apex mathematics society of France. He has been secretary of the CNRS and has acted as a consultant to the French Ministry of Education. His father Claude was a famous classical pianist and his mother Mireille an authority on Tibetan music who has done fieldwork in Ladakh. Bernard is an outstanding mathematician and I enjoyed working on our paper some years ago.

About Bernard

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

An interview with his mother Mireille Helffer

https://www.youtube.com/watch?v=DdcsxihL48Y

Our paper written in 2005

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

My paper which has been accepted and will be part of the festschrift for Bernard:

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

[Vayutuvan]

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

Academic grandson of Laurent Schwartz, Field Medalist who invented (discovered?) Generalized Functions.

[Vayutuvan]

It is already Pi day in India. So here is a YT video which is fascinating. It is both math and physics - (almost) table top physics. Not big humongous CERN type physics, not moonshot physics.

(I am off of that yoogly politicized Renewable Energy dhaaga and Understanding the US dhaaga)

Stand-up Maths
We calculated pi with colliding blocks

[Cyrano]

Fun stuff!

[Amber G.]

Happy Pi Day! - pi = 3.1416 ( APPROXIMATION ) .. sharing a page .. Aryabhatta - 471

*** Just to put in perspective:
Aryabhatta - this value of pi is not some small curiosity or his only work - His trigonometric tables for sin, cos etc are precise up to 4-5 digits. Of course, if you know, say sin of a small angle, calculating the value of pi is child's play.
The book, a picture is attached below, is a fascinating to read - one must know Sanskrit and Mathematics to really enjoy and appreciate. Unfortunately there are very few good translations.

[Amber G.]

Sharing: I just saw some old posts here in this dhaga.. about 3k+1 (also called Collatz ) problem.

For example here..
(I am copying for easy reference:

^^Thanks for posting, yes we have talked about this here.. (also known as 3k+1 problem).
Basically it is easy to state (and has been a basis for a few math problems)
- start with any number.
- If it is even divide by half
- If it is odd multiply be 3 and add 1

Repeat above with the new number you got.. what will happen? ( For example if you start with 7 it goes ->22 -> 11->34 ->17 ->52 ->26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 ->1 and then repeats 4,2,1 ...). Does this happen all the time?

This simple looking problem is still unsolved!

Recently Jeffrey Lagarias (Harold Mead Stark Distinguished Professor at the University of Michigan, Ann Arbor) delivered the Ramanujan Colloquium and two seminars at the University of Florida. Professor George Andrews sponsored this Colloquium and seminars (he has done every year since 2007). Lagarias delivered the Number Theory Seminar on the famous Collatz Problem (still unsolved) and followed this with a talk in the Combinatorics Seminar on Generalized Factorials. ..

[Picture Credit: SM of Friend]

[Hriday]

Watch the 2.22 minutes video in the link for details.

https://x.com/HPhobiaWatch/status/19020 ... 23fUw&s=19

Indian student Divya Tyagi at Penn State University has cracked a 100-year-old math problem, which will enable higher efficiency in wind turbines.

[Amber G.]

^^^Thanks.
More details from UPenn here:
Student refines 100-year-old math problem, expanding wind energy possibilities
The paper can be found at : Glauert's optimum rotor disk revisited – a calculus of variations solution and exact integrals for thrust and bending moment coefficients
----

FWIW Some comments (for those who are mathematically inclined):


The above paper "Glauert's optimum rotor disk revisited – a calculus of variations solution and exact integrals for thrust and bending moment coefficients" presents a revised solution to Glauert's optimum rotor disk problem, a fundamental concept in wind turbine design .

From a physicist's perspective, the problem involves optimizing the power coefficient of a wind turbine by finding the ideal relationship between the axial and angular induction factors. The authors use calculus of variations to derive an exact solution, which provides a more elegant and simpler mathematical approach compared to Glauert's original work .
It also explores the asymptotic behavior of the induction factors and derives exact integrals for the thrust and bending moment coefficients.

Mathematically , the paper demonstrates a beautiful application of 'calculus of variations' (A technique often used in many physics problems) to a real-world problem. The authors' use of mathematical techniques to derive exact solutions and explore the asymptotic behavior of the system is impressive and provides valuable insights into the underlying physics .

Overall, the paper presents a significant contribution to the field of wind energy and provides a valuable resource for researchers and engineers working on wind turbine design.

[Vayutuvan]

If anybody is interested to dig into Calculus of Variations, I have two books in my shelf which I have read in sections based on demand.

- The Variational Principles of Mechanics by Cornelius Lanczos https://www.amazon.com/Variational-Prin ... 486650677/.
Rigorous but very readable. He goes into a little bit into a few philosophical questions too, but only in the first chapter. Lanczos is very famous for Lanczos Iterative Algorithm for finding Eigenvalues of a matrix. He worked with and a post-doc with Einstein for a couple of years.

- Calculus of Variations by I. M. Gelfand and S. V. Fomin https://www.amazon.com/Calculus-Variati ... 486414485/

This is a very brisk book on the subject. As is the case with all dover books, it is a little harder to read. The treatment is more mathematical rather than from Classical Mechanical (i.e. physics).

Probably Lanczos is the goto book for CFD, CAE/Finite Elements folks.

[gakakkad]

there were some really good classical mechanics texts that i used to learn lagrangian/hamiltonian etc. I found those much more useful for calc. of variations than math textbooks covering the same. i personally think that the reason why lagrangian/hamiltonian etc is taught in the second physics course and not even introduced in the first course is purely for historic reasons. i think if atleast the lagrangian was introduced in Mechanics 101 or to high school kids even it 'll make them love mechanics more and reduce torture of some Pulley problems etc. Folks may remember Irodovs problem book from back in JEE days. A lot of the hard mechanics problems become a joke if you use variations . Kind of similar to how tough definite integrals with special tricks asked in desi entrance exams become a joke once you learn complex analysis .
I did ecounter Israel Gelfands book before and it's very typical russian book. very too the point . lot of things are considered trivial which are not so obvious for lesser mortals. the problems were phenomenally good though not a ton of them.

[Kanoji]

Happy Pi Day! - pi = 3.1416 ( APPROXIMATION ) .. sharing a page .. Aryabhatta - 471

*** Just to put in perspective:
Aryabhatta - this value of pi is not some small curiosity or his only work - His trigonometric tables for sin, cos etc are precise up to 4-5 digits. Of course, if you know, say sin of a small angle, calculating the value of pi is child's play.
The book, a picture is attached below, is a fascinating to read - one must know Sanskrit and Mathematics to really enjoy and appreciate. Unfortunately there are very few good translations.

Amber G sir,
Is this a page from the book "Indian Mathematics and Astronomy" by Dr Balachandra Rao?

[Amber G.]

^^^ Yes, indeed. If interested, I think ..IITK library has an online version

(It does also have "त्रिभुजस्य फलशरीरं समदलकोटी भुजार्ध संवर्गः" ( a triangle area is half the side multiplied by the perpendicular type of formulas too .. and sin/cosine tables calculated etc..Pretty impressive.

[Kanoji]

^^^ Yes, indeed. If interested, I think ..IITK library has an online version

(It does also have "त्रिभुजस्य फलशरीरं समदलकोटी भुजार्ध संवर्गः" ( a triangle area is half the side multiplied by the perpendicular type of formulas too .. and sin/cosine tables calculated etc..Pretty impressive.

Thank you.

[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!

Almost three months have passed, and the challenge remains unsolved. Despite allowing the use of AI tools, computers, calculators, internet research, and even using group resources - consultation with each other and with your favorite math professors, not a single serious attempt has been made to solve the problem.

Grok 3's attempt was, shall we say, less than impressive and almost laughable from a mathematical standpoint. (see posts above)

The challenge remains open:

Find two natural numbers (positive integers) a and b such that:
a³ + b³ = 313² × c³
where c is also a natural number.

Come on, BRF! Let's show that we can rise to the challenge. Can anyone solve this problem and claim victory?

[Vayutuvan]

I tried to plot the equation in the euclidean plane. It is a flat. It is a constrained case of fermat's equation for n=3. There may be an easy way of showing that there are no integer solutions for a, b, c > 0. 313 is a prime. variable c^3 has a coefficient which is a square of a prime. I worked on it for 30 minutes a few days back. But didn't make much progress.

[gakakkad]

^
Amberg that was a interesting question

Basically the question is A^n + B^n = kC^n with n= 3.

Now we know from fermatwa's last theorem that if k= 1 and n>2 there won't be any natural numbers satisfying that condition .

In your case k = 313^2 , n=3

I wonder if there is an integer solution there .

[Vayutuvan]

This case may be much easier to prove than the n=3 case because of the coefficient of c^3 is a square of prime number. In fact it is a Pythagorean prime.

[gakakkad]

My thoughts exactly .

We can see if general case for pythogorean primes is provable.

All pythogorean primes are in 4k+1 .

A^3 + B ^3 = (4k+1)^2 * C^3

[Amber G.]

But...(if you can see old messages here in Math dhaga) one can see 17^3+27^3 = 6*21^3
or 4^3+5^3 = 7*3^3 .....

or 2^3+1^3 = 3^2*(1^3)
or (8^3)-7^3 = 13^2*(1^3)

Of course, use a simple computer program and one can find positive integer solutions for x^3+y^3 = 13^2*z^3

Hint: This kind of problems have been discussed in BRF dhaga and in standard number theory books. (and any good math prof can give you a hint)
#nath is fun!

[Amber G.]

We can see if general case for pythogorean primes is provable.
All pythogorean primes are in 4k+1 .
A^3 + B ^3 = (4k+1)^2 * C^3

Gakakkadji, just for fun, here's a very simple problem. Maybe the technique isn't covered in elementary math, but the problem isn't hard for high school level and was described by ancient Indian mathematicians. (Similar problesm hae appeared in JEE or other math competition)

I'm putting it here because it brings out the love of math.

Can one find the integer solutions for

A^2 + B^2 = (313^2)C^2?
(Yes problem is lttle different using 2 vs 3 )

I'll leave the problem here for fun. If no one gives a solution, I'll provide a solution for solving that kind of equation.
Interestingl if 4k+1 is a prime, there is alwas a solution (oterwise for 4k-1 - Not)

[gakakkad]

^ that's easy .

A and b will be Any pythogorean triplet multiplied by 312.
C will be the "hypotenuse"

Eg a = 3*312
B= 4*312
C= 5

Or 5,12,13

Etc

We could get infinite number of natural numbers

[Amber G.]

^^^ . yes, too easy... (especially with the way I presented .. )
I meant 𝐴²+𝐵²=313²
(Which is also quite easy.... assuming none of the values is zero)

--- Speaking of Pythagorean triangles (where are sides are integers):
Few classic problems:

- Can you find a Pythagorean triangle such that:
Problem A : It's area is also perfect square
Problem B: Each leg is a perfect square. (Legs - means two sides excluding hypotenuse)
Problem C : The difference between two legs is 1 (eg 3,4 ) - Other values?
Problem D: Find a triangle (not necessary a Pythagorean) where area = perimeter.

[Vayutuvan]

Two Wikipedia pages are a hint for solving the above problem follow:

https://en.wikipedia.org/wiki/Fermat%27 ... wo_squares
https://en.wikipedia.org/wiki/Brahmagup ... i_identity

Folks can have a go at it now.

[Vayutuvan]

Generalization of the Brahmagupta-Fibonocci Identity is below.

https://en.wikipedia.org/wiki/Cauchy%E2 ... et_formula

Of interest is the continuous case called

https://en.wikipedia.org/wiki/Cauchy%E2 ... us_version

A continuous version of the Cauchy–Binet formula, known as the Andréief-Heine identity[2] or Andréief identity appears commonly in random matrix theory.[3] It is stated as follows: ...

For random matrices the standard reference is

Mehta, M.L. (2004). Random Matrices (3rd ed.). Amsterdam: Elsevier/Academic Press. ISBN 0-12-088409-7.

There is a downloadable PDF by Terence Tao

Tao, Terence (2012). Topics in random matrix theory (PDF). Graduate Studies in Mathematics. Vol. 132. Providence, RI: American Mathematical Society. p. 253. doi:10.1090/gsm/132. ISBN 978-0-8218-7430-1

https://terrytao.files.wordpress.com/20 ... x-book.pdf

[Amber G.]

The problems I mention exploring here are timeless classics — well-known, not too difficult for a high school student, yet still thought-provoking and fun. These are the kinds of problems that make math interesting.

They spark curiosity, nurture a love for the subject in young minds, and sometimes even open doors to deeper, more advanced mathematics.

It’s not just about solving problems — it’s about discovering the joy and beauty that math has to offer.

Here are the typical ways - solutions to the problems — not just answers, but approaches I encourage others to try, reflect on etc.. There’s more than one path through a good math problem, and that’s part of what makes it so rewarding.

For 𝐴²+𝐵²=313²

(Which is also quite easy.... assuming none of the values is zero)

The method is given by Brhamgupta - here is modern methd"
313 = 13² +12². (I gave that property - as said

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.

So 313 =(13+12i)(13-12i) so
313² = ((13+12i)²)((13-12i)²)
= (25+312i)(25-312i) = 25² +312²

(This is one of the use of Brahmkguta's study of sum of two squared. we do complex algebra - and some time study Gaussian primes.)

(Note: From Number theory: Each prime of the type 4k+1 can be written only one way as sum of two squares. - Reverse is also true - if there is one and one way to write is sum of two squares then the number is prime. Primes 4k-1 do not share this property)

For those who are curious:
Problem A - No solution (Not too hard to prove)
Problem B - No solution
Problem C - Infinte solutions (eg 20,21,29)
Problem D - I will wait ..)

[Vayutuvan]

Gian-Carlo Rota

Mathematicians also make terrible salesmen. Physicists can discover the same thing as a mathematician and say "We've discovered a great new law of nature. Give us a billion dollars." And if it doesn't change the world, then they say, "There's an even deeper thing. Give us another billion dollars."

https://fs.blog/gian-carlo-rota/

Don’t worry about small mistakes

“Once more let me begin with Hilbert. When the Germans were planning to publish Hilbert’s collected papers and to present him with a set on the occasion of one of his later birthdays, they realized that they could not publish the papers in their original versions because they were full of errors, some of them quite serious. Thereupon they hired a young unemployed mathematician, Olga Taussky-Todd, to go over Hilbert’s papers and correct all mistakes. Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems. . . . At last, on Hilbert’s birthday, a freshly printed set of Hilbert’s collected papers was presented to the Geheimrat. Hilbert leafed through them carefully and did not notice anything.”

Rota goes on to say: “There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent ones, which are useful in testing the stability of a theory.”

Mistakes are either contingent or fatal. Contingent mistakes don’t completely ruin what you’re working on; fatal ones do. Building in a margin of safety (such as having a bit more time or funding that you expect to need) turns many fatal mistakes into contingent ones.

Contingent mistakes can even be useful. When details change, but the underlying theory is still sound, you know which details not to sweat.

Use Feynman’s method for solving problems

“Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: ‘How did he do it? He must be a genius!’”

(More such stuff at that link).

[Amber G.]

Just realized, Nicușor Daniel Dan , Romania's new president won IMO gold medals in two years and perfect scores. Probably a first.

[Amber G.]

Meanwhile: India’s IMO (International Math Olympiad) 2025 Team:
(IMO 2025 will take place in Australia July 10-20)

The official list of team members representing India at IMO 2025 has not been announced. The selection process involves the International Mathematical Olympiad Training Camp (IMOTC), organized by the Homi Bhabha Centre for Science Education (HBCSE). The IMOTC 2025 is held at the Chennai Mathematical Institute.

Looks like Ashwat Prasanna, Aarav Goel, and Sreemoyee Bera are on the team ..(among the top contenders for the final team selection).. congrats.