Scherk's theorem (number theory)

The set of Scherk is a theorem of elementary number theory , one of the part areas of mathematics . It goes back to the German mathematician and astronomer Heinrich Ferdinand Scherk . The theorem deals with the question of the representability of each prime number by means of addition and subtraction of the preceding prime numbers.

formulation

The sentence is as follows:

If the prime number sequence with designated, then: ${\displaystyle (p_{n})_{n\in \mathbb {N} }}$

(I) Every prime number that has an even index in the prime number sequence can be obtained from all smaller prime numbers as well as     from simple addition and subtraction, whereby each smaller prime number is taken into account exactly once. ${\displaystyle 1}$

(II) Every prime number, which has an odd index in the prime number sequence , can be obtained from all smaller prime numbers as well as     through simple addition and subtraction, whereby every smaller prime number is taken into account exactly once - with the exception of the next lower prime number, which is taken into account exactly twice . ${\displaystyle 1}$

In detail you have:

(I) For an even index the shape     is   always with a suitable choice of   ${\displaystyle i=2n}$     ${\displaystyle (n\in \mathbb {N} )}$ ${\displaystyle \pm }$

  ${\displaystyle p_{i}=1\pm p_{1}\pm p_{2}\pm \cdots \pm p_{2n-2}+p_{2n-1}}$   .

(II) For an odd index the shape     is   always with a suitable choice of   ${\displaystyle i=2n+1}$     ${\displaystyle (n\in \mathbb {N} )}$ ${\displaystyle \pm }$

  ${\displaystyle p_{i}=1\pm p_{1}\pm p_{2}\pm \cdots \pm p_{2n-1}+2\cdot p_{2n}}$  .

Examples

${\displaystyle {\begin{aligned}&p_{1}&=2&=2\cdot 1\\&p_{2}&=3&=1+2\\&p_{3}&=5&=1-2+2\cdot 3\\&p_{4}&=7&=1-2+3+5\\&p_{5}&=11&=1-2+3-5+2\cdot 7\\&p_{6}&=13&=1+2-3-5+7+11\\&p_{7}&=17&=1+2-3-5+7-11+2\cdot 13\\&...........\\&p_{17}&=59&=1+2-3-5+7-11+13+17-19-23+29+31-37+41-43-47+2\cdot 53\\\end{aligned}}}$

History of the sentence

Heinrich Ferdinand Scherk carried out his representation formula in the context of a heuristic view, but gave no mathematical proof for it . The first proof of the theorem was then presented by SS Pillai about a century later in 1928 . At the beginning of the 1950s, Wacław Sierpiński and E. Teuffel, independently of one another and using a similar proof approach, gave the proof that then found its way into the specialist literature via the Elementary Theory of Numbers by Sierpiński . As both proofs show, the Scherk's theorem is essentially based on the fact that when a prime number is doubled, the next largest prime number in the prime number sequence is always exceeded .

See also : Estimation (1a)

Related result: Hans-Egon Richert's theorem

A theorem related to Scherk's theorem goes back to the German mathematician Hans-Egon Richert, which deals with the sums representation of natural numbers using prime numbers in general.

The set of Richert is:

Starting with the natural number     , every natural number can be represented as the sum of unequal prime numbers. ${\displaystyle 7}$

literature

• SS Pillai : On some empirical theorem of Scherk . In: J. Indian Math. Soc . tape 17 (1927-28) , pp. 164-171 .
• Hans-Egon Richert : About decays into unequal prime numbers . In: Math. Z . tape 52 , 1949, pp. 342-343 ( DigiDok ). MR0033856
• HF Scherk : Remarks on the formation of prime numbers from one another . In: J. Reine Angew. Math . tape 10 , 1833, p. 201-208 ( DigiDok ).
• 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 ).
• W. Sierpiński : Sur une propriété des nombres premiers . In: Bull. Soc. Roy. Sci. Liège . tape 21 , 1952, pp. 537-539 ( MR0056012 ).
• E. Teuffel: Proofs for two theorems by HF Scherk about prime numbers . In: Jber. German. Math. Association . tape 58 , 1955, pp. 43-44 ( DigiDok ). MR0074448

References and footnotes

1. ↑ HF Scherk: Remarks on the formation of prime numbers from one another. in: J. Reine Angew. Math, Vol. 10, pp. 201 ff
2. ^ Wacław Sierpiński: Elementary Theory of Numbers. , Pp. 148-151
3. ↑ This is     the borderline case that, strictly speaking, is no longer covered by the above formula. ${\displaystyle p_{1}=2=2\cdot 1}$
4. ↑ Richert: Math. Z . tape 52 , p. 342-343 .
5. ↑ Sierpiński, pp. 151–153.