# Questions tagged [number-theory]

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

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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 ...
• 776
1 vote
108 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 ...
• 1,563
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### 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. ...
• 355
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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 ...
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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 ...
• 545
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 ...
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### Did Hardy and Ramanujan miscalculate these values?

When I read Dickson's History Of The Theory Of Numbers Vol-2, I found that there seems to be a mistake in the approximation of partition numbers p(200). For this reason, I found the original text ...
• 161
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 ...
• 315
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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. ...
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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 ...
• 355
139 views

### Reference for Euler's Introductio in Analysin Infinitorum

In the following answer it has been claimed that "The reference here is not to Euler's 1737 "factorization" of the harmonic series but to 1748 Introductio in Analysin Infinitorum, where the identity ...
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309 views

### 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 ...
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### What is the intuition behind Brahmagupta’s rule for multiplying negative numbers?

The rule says: The product (or quotient) of two debts is a fortune What I’m struggling with is what exactly is the product of two debts? What accounting need forces one to multiply debts? How do ...
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