Bharat Rakshak

To add about Prime number theorem - Gauss mentioned and studied it, but it was not proven till almost 1900 (Poussin ? ) (using very sophisticated/complex methods) Erdos (my hero as mentioned in previous pages) was the first one to give without using complex variables and, in my opinion, is the only proof I have seen which is easy to understand..

Ramanujan, of course, is famous for tackling some of the most complex results/improvement of this.. (His original claim to find the precise formula did not come out to be correct but he did refined these results quite a bit)

But we may be getting to much out of the general discussion..here ...

BTW have you read the book "Men who knew Infinity" .. if not, strongly recommend it. (This is a book about Ramanujan).

Dr. Hobbs got accepted to Waterloo, Paul Erdos also visited Waterloo since he dabbled in graph theory (as with everything) himself. Hobbs found out that Erdos was going to some conference and made sure to get on the same bus. He went up to Erdos and expressed his desire to publish a paper with him (he wanted an Erdos number of 1). So, Erdos asked him if he had any idea for what they could write about. Hobbs replied, "I was hoping you did." Erdos sighed and asked Hobbs what he did. "I did my thesis on Hamiltonian cycles." Okay... is every cubic graph with girth less than 4 Hamiltonian? Hobbs thought he would try the Dirac formula on the question, but if Erdos hadn't solved it yet then that probably didn't work. Sure enough, the Dirac formula didn't work. He woke up Erdos (Erdos tended to sleep when people bored him) and told him what he found. Erdos nodded, gave him an idea of a proof, and where the hard spots were. Hobbs went back to work, solved the problem, and got his Erdos number

Also "The Man Who Loved Only Numbers." It's by Paul Hoffman.

^^^ Thanks about mentioning Dr Hobbs. Small world. He was a good friend of Erdos and liked to tell stories about him in his class ..there is interesting story told by him (I am sure you have heard it) of becoming Erods number of 1 (those who wrote a paper with him)

Here is the story..(Thanks to google I found a reference so I can just cut and paste)

His precocious fascination for epic poetry began
with Homer’s Iliad, in Greek, which in turn quickened
his interest in Sanskrit, and led inevitably to
the great epic Maha¯bha¯rata. Its core, the Bhagavad-
Gõ¯ta (the Song of God) stirred his blood as
nothing else did either before
or since. He acquired
sufficient Sanskrit to be
able to read the Gõ¯ta in the
original with the help of a
Sanskrit-French dictionary
and an English translation.
He was taken in as much
by the beauty of the poem
as by the thought that inspired
it. The Gõ¯ta is perhaps
the most systematic
spiritual statement of the
“perennial philosophy”,
embodying those universal
truths to which no one
people or age can make exclusive
claim. Eminent Indologists
like A. K.
Coomaraswamy have expressed
the opinion that
it is “probably the most
important single work produced
in India.” The Gõ¯ta remained Weil’s close companion
all his life, through thick and thin, as it did
with Gandhi.
Ka¯lida¯sa’s lyric poem Meghadu¯ta (the Cloud-messenger)
enraptured him—as it has at least fifty generations
of Indians—with its delicacy and grace, its
mellifluous diction, its lyrical concision, and its suggestive
power. Ka¯lida¯sa’s deceptively simple Sanskrit
is the despair of translators. Weil was struck by
Ka¯lida¯sa’s mastery of the Sanskrit language, of its
grammar and rhetoric and dramatic theory, “subjects
which Hindu savants have treated with great, if sometimes
hair-splitting, ingenuity,” in the words of Arthur
Ryder (Berkeley, 1912). He came upon the fact that
in India, Pa¯n. ini’s invention of grammar (ca. fourth
century B.C.) had preceded that of the decimal notation
and negative numbers. Pa¯n. ini’s As.t.
a¯dhya¯yi
(eight chapters) consists of nearly four thousand
aphorisms, the su¯tras, enumerating the technical
terms used in grammar and the rules for their interpretation
and application. The Sanskrit term for
grammar is Vya¯karan.a, which literally means “undoing”,
implying linguistic analysis. Weil could very
well say that “nothing he later came across in the writings
of Chomsky and his disciples seemed unfamiliar
to him.”
Having delved that deep into Sanskrit studies, he
was ready to jump at any offer of a chair in India,
which eventually turned out to be mathematics
(1930–32). It was in Helsinki on the opening day of
the Congress (ICM 1978), as we emerged from a reception
given by our Finnish hosts, and the evening
was spread against the sky, that he suddenly asked
me to recite the first line of the first stanza of
Meghadu¯ta, which he so dearly loved. I had not then
known, as I did later, the intensity of the impact of
India on his personality. Joseph Brodsky has said: “A man is what he
loves. That is why
he loves it; because
he is a part
of it.” His colleagues
at Princeton,
Gödel and
Oppenheimer,
were ardent admirers
of the Gõ ¯ta.
Oppenheimer’s citation
of lines
(often misquoted)
from the Gõ ¯ta (Ch.
XI, verse 32) as he
witnessed the
first nuclear explosion
has entered
the history
of American science.
Neither of
them, however,
could visit India as they had wished

The aura of Srinivasa Ramanujan, perhaps one of the greatest mathematicians of all time, has virtually eclipsed other Indian mathematicians

One remarkable fact about transcendental numbers are that they are uncountable, although they seem to be only few (e, pi etc.).

Dear Mrs. Feynman,
We have not met, I believe, frequently enough for either of us to have taken root in the other's conscious memory. So please forgive any impertinence, but I could not let Richard's death pass unnoticed, or to take the opportunity to add my own sense of loss to yours.
Dick was the best and favorite of several “uncles” who encircled my childhood. During his time at Cornell he was a frequent and always welcome visitor at our house, one who could be counted on to take time out from conversations with my parents and other adults to lavish attention on the children. He was at once a great player of games with us and a teacher even then who opened our eyes to the world around us.
My favorite memory of all is of sitting as an eight-or-nine-year-old between Dick and my mother, waiting for the distinguished naturalist Konrad Lorenz to give a lecture. I was itchy and impatient, as all young are when asked to sit still, when Dick turned to me and said, “Did you know that there are twice as many numbers as numbers?”
“No, there are not!” I was defensive as all young of my knowledge.
“Yes there are; I'll show you. Name a number.”
“One million.” A big number to start.
“Two million.”
“Twenty-seven.”
“Fifty-four.”
I named about ten more numbers and each time Dick named the number twice as big. Light dawned. {75}
“I see; so there are three times as many numbers as numbers.”
“Prove it,” said Uncle Dick. He named a number. I named one three times as big. He tried another. I did it again. Again.
He named a number too complicated for me to multiply in my head. “Three times that,” I said.
“So, is there a biggest number?” he asked.
“No,” I replied. “Because for every number, there is one twice as big, one three times as big. There is even one a million times as big.”
“Right, and that concept of increase without limit, of no biggest number, is called ‘infinity.’ “
At that point Lorenz arrived, so we stopped to listen to him.
I did not see Dick often after he left Cornell. But he left me with bright memories, infinity, and new ways of learning about the world. I loved him dearly.

Historically, did indians excel more than the chinese in mathematics (discoveries, styles, etc)? ...

...Go seems more complicated and takes a bit more creativity to beat the opponent vs chess.

रानी बढ़ी कालपी आई, कर सौ मील निरंतर पार,
घोड़ा थक कर गिरा भूमि पर गया स्वर्ग तत्काल सिधार,
यमुना तट पर अंग्रेज़ों ने फिर खाई रानी से हार,
विजयी रानी आगे चल दी, किया ग्वालियर पर अधिकार।


अंग्रेज़ों के मित्र सिंधिया ने छोड़ी रजधानी थी,
बुंदेले हरबोलों के मुँह हमने सुनी कहानी थी,
खूब लड़ी मर्दानी वह तो झाँसी वाली रानी थी।।

^^^ Historic reason for the fort is captured in this

Gwalior Fort also occupies a unique place in the human civilization as the place which has the first ever recorded use of zero. Also referred as 'Shunya' in sanskrit, this site is of mathematical interest because of what is written on a tablet recording the establishment of a small 9th century Hindu temple on the eastern side of the plateau. By accident, it records the oldest "0" { see "270 in picture above} in India for which a definite date can be assigned

^^^ Here is another cute proof .. (Think attributed to Goldbach)
Take F_n = 2^(2^n)+1 (Known as Fermat's numbers)
Easy to prove, there are no common factors between any two F_n's
Hence, each F_n has (at least) one new prime number as its factor ..
Since there are infinite F_n's (n can become as large as one wants).. there are infinite prime numbers..

Proof by contradiction:
If the number is finite ... Let p be the largest prime.
Take the number 11111... (p times)
Any factor of the above is larger than p.. (why? - Can you prove this)
Contradiction...
QED

OK. The other one using Lagrange's Theorem is similar to this. It uses coset counting.

I thought Lagrange's theorem is about the order of subgroups and the order of a group - that it essentially divides. I vaguely remember Herstein using normal subgroups (and the cosets) to arrive at the result.

Amber Ji have you read "Proofs from the book"? Lives up to the name. For others who have not read "The man who only loved numbers" (Bio of Erdos), Erdos believed that god keeps a book in which all the elegant proofs are written. Paraphrasing here - "Even if one doesn't believe in god, one should believe in the book". In that spirit the book "Proofs from the book" is written collecting some of the most elegant proofs.

Okay .. I see using it on multiplicative group(s)..
But one can achieve the same with using elementary school math using Fermat's little theorem..

On a side note, Rahul M ji note that memorization of definitions and theorems is important in certain contexts in that one doesn't prove everything starting from fundamentals. Of course, it helps to read the proof of a theorem which once internalized helps to remember the theorem itself.

My mathematical education extends up to multivariate calculus and linear algebra.

^^^ Here is another cute proof .. (Think attributed to Goldbach)
Take F_n = 2^(2^n)+1 (Known as Fermat's numbers)
Easy to prove, there are no common factors between any two F_n's
Hence, each F_n has (at least) one new prime number as its factor ..
Since there are infinite F_n's (n can become as large as one wants).. there are infinite prime numbers..

^^ Nice
(Can we ask brf admins to make brf LaTex enabled? )

I believer this is the Indian team for the 53rd International Mathematical Olympiad (2012). ( July 4-16 in Mar del Plata, Argentina)
Good luck to The Indian Team:

(1) Debdyuti Banerjee
(2) Akashdeep Dey
(3) Prafulla Sushil Dhariwal
(4) Shubham Bhakta
(5) Rijul Saini
(6) Mrudul Madhav Thatte