# Questions tagged [number-theory]

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

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### Implications of several unpublished "binomial congruences" of Gauss

In article 23 of his first memoir on biquadratic residues, Gauss gave the first example of a binomial-type congruence, which is apparently a quite "deep" result - many authors relate it to ...
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### Confusion on the original article by Lucas

I am currently researching on all primality tests deriving from Lucas' original paper Théorie des Fonctions Numériques Simplement Périodiques, which is of course known for its great deal of confusion. ...
3k views

### Did ancient Greek mathematicians consider numbers independently of geometry?

I am currently reading Oliver Bryne's edition of Euclid's Elements, and in The Elements many arithmetic propositions are proved geometrically, and it feels to me that numbers are always treated as ...
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### Who first proved that the existence of a Euclidean algorithm implies unique factorization?

In Simachew's "A Survey on Euclidean Number Fields", he said that Gauss used the existence of a Euclidean algorithm in Gaussian integers to prove that it has unique factorization. Also, he ...
109 views

### First motivation for extending Riemann Zeta to complex domain?

Euler developed the Euler Product Formula which shows that the Riemann zeta function encodes information about the prime. $$\zeta(s)=\sum_{n}\frac{1}{n^{s}}=\prod_{p}(1-\frac{1}{p^{s}})^{-1}$$ Riemann ...
1 vote
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### Was Cramér the first to interpret the PNT's $1/\log(x)$ as probability of primes?

The Cramér probabilistic model of primes is built on the assumption that the probability of $n$ being prime is $$\Pr(n)=\frac{1}{\log (n)}$$ This is not a big leap from the Prime Number Theorem which ...
55 views

### Any historical work on the distribution of prime gaps?

I am looking to see whether historic mathematicians did any work to explain the slightly unexpected distribution of prime gaps? I would have expected Gauss, who studied lists of primes and proposed a ...
1 vote
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### Euler's proof of infinite primes first since Euclid?

Q. Is it true that Euler's proof of infinite primes was the first since Euclid's which was from around 300BC? Note: By Euler's proof, I mean the use of his Euler product formula for the zeta function ...
4k views

### Is the story about Fermat's writing on a margin true?

Is there any evidence that Fermat wrote on the margin of a book "I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." Everyone ...
332 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 ...
117 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 ...
1 vote
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### 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 ...
101 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 ...
1 vote
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### 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. ...
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 ...