Questions tagged [number-theory]

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

19 questions with no upvoted or accepted answers
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What is the modern significance of Theaetetus's classification of quadratic irrationals?

Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which Fowler calls anthyphairesis. Theaetetus is said to develop a classification of ...
6
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134 views

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 ...
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277 views

What is Hensel's lemma a lemma for?

Was Hensel's lemma originally used for proving some other theorem? Or is it meant to be a standalone result? Why is it a "lemma" and not a theorem?
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98 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 ...
3
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290 views

Are there primitive Pythagoras triplets (in integers), being with all the terms as powerful numbers?

I'm searching a trusted historical sources about primitive Pythagoras triplets as being powerful integers (numerical examples), or a notable work of impossibility of such a triples, but couldn't find ...
2
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63 views

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 ...
2
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71 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 ...
2
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87 views

Notation $n=efr$ in algebraic number theory

When $\Bbb Q \subset K$ is a field extension of finite degree and when $p \in \Bbb Z$ is a prime number, the ideal $p O_K$ decomposes uniquely as a product $\prod_{i=1}^r P_i^{e_i}$ of prime ideals of ...
2
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89 views

Which is the first reference using the terminology "Chinese Remainder Theorem" for this theorem?

The Chinese Remainder Theorem is one of the fundamental theorems in modular arithmetic. As far as I know, this terminology for the theorem is due to the fact that the Chinese mathematicians were the ...
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99 views

How did Fermat come up with his Last Theorem?

It's usually believed that Fermat's claim that he had a proof for the Last Theorem is false, and that it might have been more of a conjecture. Or considering it took many centuries and advanced ...
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103 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 ...
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35 views

Timeline for the earliest work on Frobenius problems

If $a, b$ are positive and coprime integers, then the set of linear combinations of $a$ and $b$ with nonnegative coefficients is all integers past $(a - 1)(b - 1)$; i.e. $\{ \lambda_1 a + \lambda_2 b :...
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92 views

History of Reciprocity Laws

Does anybody know a freely available overview of the history of Reciprocity Laws, especially the Cubic and Biquadratic ones? Wikipedia I know about Franz Lemmermeyer's book "Reciprocity Laws", but ...
1
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1answer
110 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 ...
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50 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 ...
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98 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 ...
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139 views

Why isn't François Proth's name used for Gilbreath's conjecture as he discovered & published a proof 80 years earlier?

According to Wikipedia's Gilbreath's conjecture page, The statement is named after mathematician Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the ...
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78 views

How did Hardy and Littlewood formulate the k-tuple conjecture?

Let $\mathcal{H}_k = (h_1,h_2,\cdots,h_k)$ be an admissible k-tuple. The k-tuple conjecture predicts that the number of primes $(b+h_1,b+h_2,\cdots,b+h_k)\in \mathbb{P}^k$ with $b+h_k \leq x$ is: $$\...
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109 views

Was 360 considered a magic number, possibly?

The number $360$ as the number of units into which the circle is divided has some nice properties: it has as many divisors as a number of its size can have it's nearly the number of days per year ...