Erdős theorem (number theory)

The set of Erdős is a theorem of number theory , one of the branches of mathematics . It goes back to the important Hungarian mathematician Paul Erdős .

The theorem is related to a conjecture formulated in 1849 by the French mathematician Alphonse de Polignac (1817-1890) , which states that every odd natural number has a representation where a natural number is while a prime number is or . ${\ displaystyle n> 1}$ ${\ displaystyle n = 2 ^ {k} + p}$ ${\ displaystyle k}$ ${\ displaystyle p}$ ${\ displaystyle p = 1}$

With his theorem Erdős was able to show that Polignac's conjecture is wrong in an infinite number of cases.

formulation

The sentence can be stated as follows:

There is an infinite arithmetic sequence , which consists of all odd natural numbers , ${\ displaystyle n}$

of which “none” can be represented in the form with an integer and a prime number . ${\ displaystyle n = 2 ^ {k} + p}$ ${\ displaystyle k \ geq 0}$ ${\ displaystyle p}$

Lemma to prove it

The proof of the theorem is based on the following elementary lemma:

Every natural number always fulfills at least one of the following six congruences . ${\ displaystyle k}$

(1)

{\ displaystyle k \ equiv 0 \ mod 2}

(2)

{\ displaystyle k \ equiv 0 \ mod 3}

(3)

{\ displaystyle k \ equiv 1 \ mod 4}

(4)

{\ displaystyle k \ equiv 3 \ mod 8}

(5)

{\ displaystyle k \ equiv 7 \ mod 12}

(6)

{\ displaystyle k \ equiv 23 \ mod 24}

From this it follows that one of six further congruences must always be fulfilled, with the help of which one wins the sentence using the Chinese remainder of the sentence. ${\ displaystyle 2 ^ {k}}$

literature

Paul Erdős: On integers of the form 2 ^k + p and some related problems . In: Summa Brasiliensis Mathematicae . tape 2 , 1950, p. 113-123 ( renyi.hu [PDF]).

Wacław Sierpiński : Elementary Theory of Numbers (= North-Holland Mathematical Library . Volume 31 ). 2nd revised and expanded edition. North-Holland ( inter alia), Amsterdam ( inter alia ) 1988, ISBN 0-444-86662-0 ( MR0930670 ).

References and footnotes

↑ ^a ^b Sierpiński: p. 445.
↑ Erdős: On integers of the form 2 ^k + p and some related problems . In: Summa Brasiliensis Mathematicae . tape 2 , 1950, p. 113 .

[Sierpiński-1] Sierpiński: p. 445.

[2] Erdős: On integers of the form 2 ^k + p and some related problems . In: Summa Brasiliensis Mathematicae . tape 2 , 1950, p. 113 .