Wieferich prime number

A Wieferich prime number is a prime number with the property that it is divisible by . ${\displaystyle p}$ ${\displaystyle 2^{p-1}-1}$ ${\displaystyle p^{2}}$

Alternatively, you can write this as congruence :

${\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}.}$

Such prime numbers were first described in 1909 by the German mathematician Arthur Wieferich .

Well-known Wieferich prime numbers

So far only two Wieferich prime numbers are known, namely 1093 (Waldemar Meißner 1913) and 3511 ( Beeger 1922). With the help of a computer, all numbers up to 6.7 × 10 ^{15 were} examined until November 2008 ; no other Wieferich prime numbers were found. It is not known whether there are infinitely many Wieferich prime numbers. There is both the assumption that this is not the case and the opposite, more precisely: that between and about Wieferich prime numbers lie. It is even still open whether there is an infinite number of prime numbers that are not Wieferich prime numbers. Joseph Silverman showed this in 1988, assuming the abc conjecture . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \log(\log(y)/\log(x))}$

Relationship with the large Fermat sentence

Wieferich dealt with Fermat's great theorem . In 1909 he published the sentence as a result:

When , where and integers are a prime number and the product is not divisible by , then a wieferich prime, so by divisible. ${\displaystyle x^{p}+y^{p}+z^{p}=0}$ ${\displaystyle x,y}$ ${\displaystyle z}$ ${\displaystyle p}$ ${\displaystyle xyz}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle 2^{p-1}-1}$ ${\displaystyle p^{2}}$

In 1910 Dmitry Mirimanoff showed that it is then also divisible by . The only known prime numbers that meet this condition are and (Kloss 1965). ${\displaystyle 3^{p-1}-1}$ ${\displaystyle p^{2}}$ ${\displaystyle p=11}$ ${\displaystyle p=1006003}$

From Fermat's great theorem, proved in 1995, it follows that the requirements of Wieferich's theorem cannot be fulfilled.

Properties of Wieferich prime numbers

• The Mersenne number can be constructed as a product from the Wieferich prime number . ${\displaystyle w}$ ${\displaystyle M_{n}=M_{w-1}=2^{w-1}-1}$ ${\displaystyle M_{w-1}=kw^{2}}$

${\displaystyle n=w-1}$is thus (trivially, since it is even) not prime, and not a Mersenne prime number . ${\displaystyle n}$ ${\displaystyle M_{n}}$

• The open question is whether there are Mersenne numbers (with prime exponents ) that are divisible by. There must be a divisor of if it should be divisible by . ${\displaystyle M_{p}<M_{w-1}}$ ${\displaystyle p}$ ${\displaystyle w^{2}}$ ${\displaystyle p}$ ${\displaystyle w-1}$ ${\displaystyle M_{p}}$ ${\displaystyle w}$

This fact can be expressed in terms of group theory:

Since is not prime, it is not a Mersenne number. It would have a Mersenne number with be caused by is divisible; d. This means that the length of the multiplicative cyclic subgroup from to base 2 should be prime. ${\displaystyle w-1}$ ${\displaystyle 2^{w-1}-1}$ ${\displaystyle 2^{p}-1}$ ${\displaystyle p={\frac {w-1}{x}}}$ ${\displaystyle w^{2}}$ ${\displaystyle g(w)}$ ${\displaystyle w}$

But it is empirically the group orders of the only known Wieferich primes and not prime. ${\displaystyle g(1093)=364=4\cdot 7\cdot 13}$ ${\displaystyle g(3511)=1755=3^{3}\cdot 5\cdot 13}$

The fact that Mersenne numbers are free of squares seems to be only an empirical result so far. Mathworld formulates, for example, "All known Mersenne numbers are square-free. However, GUY (1994) suspects that there are Mersenne numbers that are not square-free" ${\displaystyle 2^{p}-1}$ .

• Difference to bases other than 2: for bases other than 2 and the corresponding equivalents to Mersenne and Wieferich numbers, this does not apply.

E.g. the condition divides (with prime) is fulfilled for base 3 . ${\displaystyle {\frac {3^{5}-1}{3-1}}=11^{2}}$ ${\displaystyle w^{2}}$ ${\displaystyle 3^{p}-1}$ ${\displaystyle w,p}$

Occurs to the base in 2819 in the Wieferich analogy even to the power of the fourth The freedom from squares of Mersenne numbers (to base 2) must therefore be a special property of base 2 (and possibly other bases) if it should apply in general. ${\displaystyle w=19}$ ${\displaystyle 2819^{3}-1=x\cdot 19^{4}}$ ${\displaystyle w=19}$

• For a Wieferich prime the following applies: ${\displaystyle p}$

${\displaystyle 2^{p^{2}}\equiv 2{\pmod {p^{2}}}.}$

• With always occurs simultaneously . ${\displaystyle 2^{n}\equiv 1\ mod\ p}$ ${\displaystyle 2^{n}\equiv 1\ mod\ p^{2}}$

literature

• Paulo Ribenboim : The world of prime numbers. Secrets and Records. Springer, Berlin a. a. 2006, ISBN 3-540-34283-4 ( Springer textbook ; updated translation of The little book of bigger primes . Springer, New York 2004)

Web links

• Wieferich @ Home - search for Wieferich prime . (English)
• Eric W. Weisstein : Wieferich Prime . In: MathWorld (English).
• Wieferich prime . at the Prime Pages of Chris K. Caldwell (English)

Individual evidence

1. ↑ ^{a b} Arthur Wieferich : On the last Fermat's theorem . In: Journal for pure and applied mathematics , 136, 1909, pp. 293-302
2. ↑ Waldemar Meißner: About the divisibility of 2 ^p −2 by the square of the prime number p = 1093 . In: Meeting reports of the Royal Prussian Academy of Sciences , July 10, 1913, pp. 663–667
3. ↑ NGWH Beeger : On a new case of the congruence 2 ^{p − 1} ≡ 1 (mod p ² ) . In: Messenger of Mathematics , 51, 1922, pp. 149–150 (English) Textarchiv - Internet Archive
4. ↑ François G. Dorais, Dominic W. Klyve: A Wieferich prime search up to 6.7 x 10 ¹⁵ . In: Journal of Integer Sequences , October 14, 16, 2011, Article 11.9.2
5. ↑ Wieferich prime . at the Prime Pages of Chris K. Caldwell (English)
6. ^ Richard Crandall , Karl Dilcher, Carl Pomerance : A search for Wieferich and Wilson primes . In: Mathematics of Computation , 66, January 1997, pp. 433–449 (English)
7. ^ Joseph H. Silverman : Wieferich's criterion and the abc-conjecture . In: Journal of Number Theory , October 30, 1988, pp. 226–237 (English)
8. ^ D. Mirimanoff : Sur le dernier théorème de Fermat . In: Comptes rendus hebdomadaires des séances de l'académie des sciences , 150, 1910, pp. 204–206, Textarchiv - Internet Archive ; extended version: Sur le dernier théorème de Fermat . In: Journal for pure and applied mathematics , 139, 1911, pp. 309–324 (French)
9. ↑ KE Kloss: Some number-theoretical calculations . In: Journal of Research of the National Bureau of Standards , 69B, October – December 1965, pp. 335–336 (English; Zentralblatt review )
10. ↑ Eric W. Weisstein : Mersenne Number . In: MathWorld (English).