# Questions tagged [number-theory]

A field of mathematics studying numbers, their properties and structures that arise from them.

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### When and where was Legendre's Conjecture first published?

When and where did Legendre first publish or write about his conjecture that there is a prime between consecutive square numbers? $$n^2 < p < (n+1)^2$$ I have looked through edition 1 and 2 of ...
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### Special quadratic reciprocity? $(-3/p)_2$ and $(5/p)_2$ in addition to $(-1/p)_2$ and $(2/p)_2$?

Context: since $\mathbb Z/p^\times$ is cyclic for $p$ prime, $-1$ is a square mod $p$ if and only if $(-1)^{{p-1\over 2}}=1$. A significantly subtler, but still classical, case is determining the ...
586 views

### Pythagorean and the number 17

I was looking for some properties of the prime numbers, and I found some curiosities of the number 17 in that site. One of them is this 17 was called by the Pythagoreans - opposition, obstruction,...
1 vote
107 views

### Number theory: a quote

I remember reading a remark on number theory that went something like this: "there is plenty of material in this topic for an $n$ semester course without having to repeat oneself". Have you ...
183 views

### Simplest of the many proofs the prime harmonic series diverges

Over the history of mathematics, some key facts have had multiple and different proofs developed for them. Sometimes these different proofs provide a unique insight or understanding of those facts. ...
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### Who first proved that there is no five-digit perfect number?

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. The first five perfect numbers are 6, 28, 496, 8128, and 33550336. Nicomachus ...
149 views

### Hilbert's problem list did not include Fermat's last theorem. Why?

Fermat's Last Theorem was open for more than 350 years until Andrew Wiles proved it in 1995. Are there possible (historical or other) reasons why David Hilbert did not include this famous open problem ...
1 vote
110 views

### Fermat et l'équation de Pell

Does any of you happen to own and electronic copy of the paper "Fermat et l'équation de Pell" by A. Weil? If I understand correctly, this paper can be found on pages 413-419 of the third ...
10k views

### How much did John Nash contribute to proving the Riemann hypothesis?

At quite the end of the movie A Beautiful Mind, John Nash tells a student "I am making progress" (towards proving the Riemann hypothesis (RH)). Actually, how much did Nash contribute to the ...
1k views

### Why is the Sophie Germain Identity called thus?

Several authors (z.B.: Arthur Engel in his Problem-Solving Strategies, Alexander Bogomolny in this entry of the Cut the Knot website) refer to the following (straightforward) consequence of the ...
163 views

### Original Statement and Proof of Chebyshev’s $\vartheta(x) \sim x$

I am looking for an online source that gives the original statement and proof of Chebyshev’s asymptotic relation $\vartheta(x)\sim x$ where $\vartheta(x)=\sum_{p\leq x}\log p$. I have tried ...
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### Who first identified $\frac{n}{\ln(n)}$ as an approximation of a prime counting function?

Gauss, in his 1849 letter to Encke, mentions that he noticed the primes have a density approx $\frac{1}{\ln(n)}$. In that letter, he also mentions an integral function for approximating the prime ...
166 views

### Has any large group of people used a base other than 10, 20 and 60 for ordinary purposes?

Wikipedia's list of numeral systems lists only $10,20,60$ as having been used in history. There are about twenty-five sets of symbols there used by different groups of people, but only three different ...
1 vote
121 views

### How were number symbols derived/shaped up?

This question was sitting on my to do list for sometime. So, as I was reading a book on history of science, I came across of a paragraph where the author attempted to give a historical development ...
560 views

### Who extended the Euclidean algorithm to derive the Bezout identity?

Euclidean algorithm is an algorithm that produces the greatest common divisor of two integers. It was described by Euclid as early as in 300 BC. On the other hand, the extended Euclidean algorithm ...
266 views

### What changes in mathematics resulted in the change of the definition of primes and exclusion of 1?

Why 1 is not prime? I read in this article that G.H Hardy explicitly included 1 as a prime in the first 6 editions of "A Course in Pure Mathematics", published between 1908-1933. He updated ...
109 views

### Where is First-Order Peano Arithmetic first clearly formulated?

I really should know this, but ... When/where/by whom was first-order Peano Arithmetic first clearly and explicitly formulated in a recognizably modern form (perhaps exact notation apart) -- with the ...
99 views

### Does this mathematical result have a specific name?

I am not sure if it's new although it may be an easy consequence of some theorem or lemma.The result is as follows: By choosing a set of numbers between $0$ and $n$(for any $n$) picking each number at ...
246 views

### What is the basis of the claim that $F_5$ was fully factored in 1732?

The Wikipedia Page on Fermat numbers states that $F_5$ was "fully factored" in 1732. This appears to be the same time that Euler found that any factor of a Fermat number $F_n$ was of the form 2^{n+...
1 vote
123 views

### Why didn't the ancient Greeks consider 1 to be odd?

The Wikipedia page on parity currently says: The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even Why didn't they consider 1 as odd? (I am assuming they already had the ...
1 vote
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### What are the direct predecessors of Lagrange's theory of quadratic forms?

I was reading Stillwell's Mathematics and its History, where Lagrange's theory of quadratic forms is synoptically presented, and I was wondering of what are the direct predecessors of the theory. ...