3
votes
1answer
88 views

History of $p$-adic numbers

I'm interested in learning about the historical motivation and development of $p$-adic numbers. I haven't been able to find any books on the topic. I'd appreciate any references, including to more ...
3
votes
1answer
70 views

A Concept Which Has Been 'Specialized' In the Course of History

There are so many concepts which have been generalized during history of mathematics - the concept of "number" may be the best examples. On the other hand, a concept may have been specialized ; the ...
8
votes
2answers
171 views

Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?

Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator? In order to say clearly, this number should given by a certain formula, such as ...
25
votes
4answers
5k views

Why are so many of the oldest unsolved problems in mathematics about number theory?

Stillwell mentions in his book, Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... Have ...
3
votes
0answers
123 views

What makes Beal's conjecture “beautiful” enough to make people offer a million dollar prize? [closed]

What makes Beal's conjecture "beautiful" enough to make people offer a million dollar prize? Is it just a challenge or does it have real applications?
1
vote
0answers
90 views

Mathematical foundation crisis and the RSA

I am currently in my last year of high school and I am writing a report on cryptography from a idea historical and mathematical perspective. I am including a few of the subjects: Cantor's diagonal ...
8
votes
4answers
558 views

Can we prove that odd and even numbers alternate without using induction?

It is a simple exercise to prove using mathematical induction that if a natural number n > 1 is not divisible by 2, then n can be written as m + m + 1 for some natural number m. (Depending on your ...
9
votes
1answer
347 views

Origin of Littlewood's idea about sign changes of $Li(x) - \pi(x)$

Background (skip if you like). Skewes and Littlewood are closely identified with the idea that $Li(x)- \pi(x)$ changes sign infinitely often, but Littlewood closed a gap in the work of Schmidt, who ...
3
votes
1answer
164 views

Calculation involving $\int_2^x \frac{dx}{\log x}$

Background (skip to the gray if you prefer). In Legendre's 1798 work on number theory he conjectured that $\pi(x)\sim \frac{x}{\log x - A}$ in which he proposed that $A = 1.08366.$ Gauss disputed the ...
3
votes
1answer
84 views

Expressing integers as a sum of squares

There have been many results about the number of squares needed to represent a positive integer. Lagrange's four-square theorem tells us that $4$ squares suffice for any integer and there have been ...
19
votes
1answer
337 views

Matsunaga's Method for solving $x^2+y^2=p$

In his history of number theory, Dickson mentions an 18th century algorithm due to Matsunago [Sic --- he means, presumably, Matsunaga Ryohitsu a.k.a. Matsunaga Yoshisuke] for finding two numbers whose ...
4
votes
0answers
169 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
3
votes
0answers
262 views

Origins of the Twin Prime Conjecture

The exciting new results by Zhang and others about bounds on the gaps between pairs of primes have been getting a fair amount of press, which is great! Some of them have gotten me wondering about the ...
5
votes
0answers
178 views

Ramanujan and sum of four cubes

This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem: Every multiple of 6 can be written as a sum of ...
30
votes
2answers
6k views

Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who ...
8
votes
4answers
506 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
10
votes
2answers
122 views

Origin of well-ordering proof of uniqueness in the FToArithmetic

In the Appendix to Ivan Niven's book "Numbers: Rational and Irrational", he proves the Fundamental Theorem of Arithmetic (FToA) without using Euclid's Lemma that if a prime divides a product, then it ...
5
votes
1answer
267 views

A question on Paul Erdős's research on Egyptian fractions

A good day to everyone! I have a (somewhat) intriguing question regarding Paul Erdős's papers on Egyptian fractions (e.g., his 2nd paper during his mathematical career was about this topic). My ...
5
votes
3answers
611 views

Is there a text that provides the proof of Fermat's Last Theorem?

I know that professor Andrew Wiles discovered his proof of Fermat's Last Theorem in 1995. One of my friends is looking for a text which provides his proof. I know that the proof is very complicated ...
-1
votes
4answers
258 views

The facts about $\varphi$ [closed]

A lot of people believe there is something special about the number $\varphi= \frac {1+ \sqrt5}{2}$. However, I can only think of cultural explanations for looking at each property of $\varphi$ as ...
4
votes
2answers
144 views

Plato's Disc of Gold

In the book Mathematical Cranks, Underwood Dudley describes the following problem on page 36: Dear Archimedes, Your problem is solved but:-- About twenty years ago he lived on Crete and was ...
3
votes
1answer
203 views

Early history of lower bounds on the prime counting function

Let $\pi (x)$ be the number of prime numbers less than or equal to $x$. Euclid's proof of the infinitude of primes gives a horrible lower bound of the type $ \pi (x)>> \sqrt{\log{x}} $. ...
3
votes
1answer
209 views

History of the study of rational points on the circle

What is the first known instance of a mathematician parameterizing rational points on the unit circle by the slopes of rational lines going through a rational point on the circle?
4
votes
1answer
122 views

Nagura's paper--can we substitute for the original upper bound?

This question concerns two results about primes. The first is J. Nagura's 1952 result, that there is a prime on the interval $[x, (1+1/5)x] $ for $x> 2103,$ which depends on the result derived ...
18
votes
0answers
452 views

Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of ...
12
votes
3answers
926 views

History of elliptic curves

In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying ...
1
vote
1answer
149 views

Which automata recognise the algebraic numbers?

I am reading historical stuff on the algebraic and transcendental numbers. Descartes, in his Geometry, excluded all curves not expressible as algebraic equations. Later, Leibniz called such curves ...
3
votes
2answers
805 views

Can somebody simply explain Wilson's theorem (for a 13 year old)

I am Rohan Kapur. This is my first time posting on the Mathematics site, although I am quite active on StackOverflow, the programming site. I am doing a Islamic Maths assignment at the moment for ...
9
votes
1answer
729 views

ancient concepts and modern concepts

Is there an extant published expository account, comprehensible to all mathematicians, of the conceptual differences between ancient Greek mathematical concepts and modern ones? I have in mind things ...
3
votes
1answer
558 views

Reference request for “Weierstrass equation” and “Weierstrass normal form”

I would like to know more about the history of the widely used terms "Weierstrass equation" and "Weierstrass normal form", as they appear in the theory of elliptic curves. When were these terms first ...
2
votes
1answer
96 views

Chebyshev/Tchebycheff's results concerning $\phi(x)$.

The book "A SOURCE BOOK IN MATHEMATICS" has a great collection of mathematical papers. On of the is Chebyshev's Memoir on "The Totality of Primes Less Than a Given Number." The book states that ...
-2
votes
1answer
276 views

Euler's Choice and Riemann's Oversight?

Euler's Choice: When Euler crafted the zeta function, he knew that $\zeta(1)$ diverged, so he made $\zeta(1)$ undefined. When he crafted the zeta generating function using the Bernoulli numbers, ...
6
votes
3answers
1k views

Proving Fermat's Last Theorem for n=3 using Euler's and Lamé's approach?

Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by ...
9
votes
3answers
531 views

Fibonacci numbers modulo $p$

If $p$ is prime, then $F_{p-\left(\frac{p}{5}\right)}\equiv 0\bmod p$, where $F_j$ is the $j$th Fibonacci number, and $\left(\frac{p}{5}\right)$ is the Jacobi symbol. Who first proved this? Is there ...
0
votes
1answer
363 views

Resurrection of my Tamagawa numbers Question, to understand the Formulation of BSD

My previous question was closed very badly for asking the broad and deep things, so I now understand the consequences of asking such questions, so I refrain from asking such questions, so this is not ...
3
votes
1answer
102 views

Counting bases to which numbers are pseudoprime

Let $n=p_1^{a_1}\cdots p_k^{a_k}$ be an odd composite. Then the number of bases $1\le b\le n-1$ for which n is a strong pseudoprime is $$ \left(1 + \frac{2^{k\nu}-1}{2^k-1}\right) ...
10
votes
2answers
375 views

What's the history of the result that $p_{n+1} < p_n^2$, and how difficult is the proof?

In Edsger Dijkstra's monograph "Notes on Structured Programming", he describes a simple imperative program for generating an array of the first $n$ primes. For each prime $p_n$, it finds the next ...
11
votes
2answers
676 views

Fermat's Last Theorem: implications (there is no new proof)

I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number ...
11
votes
1answer
372 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
43
votes
10answers
6k views

Is zero odd or even?

Some books say even numbers start from two but if you consider the number line concept, I think zero should be even because it is in between -1 and +1 (i.e in between 2 odd numbers). What is the real ...
4
votes
2answers
409 views

Politics of the Adelics

The adelics seem counter-intuitive. I wonder how they came up originally, and what was the immediate reward for introducing them. What was the politics of introducing the adelics into mathematical ...