Number of Wieferich

In Mathematical, a prime number of Wieferich is a Prime number p such as $p^2 \,$ divides $2^ {p-1} - 1 \,$ , to compare this with Small theorem of Fermat, which states that each prime number p divides $2^ {p - 1} - 1 \,$ . The prime numbers of Wieferich were described in first by Arthur Wieferich in 1909 in its work relating to the Dernier theorem of Fermat.

The research of the prime numbers of Wieferich

The only known prime numbers of Wieferich are 1093 and 3511 (electronic Encyclopédie of the whole continuations (id=A001220), found by W. Meissner in 1913 and NR. G.W.H. Beeger in 1922, respectively; so of others exist, they must be $> 1,25.10^ {15} \,$ . It was conjectured that there exists only one finished number of prime numbers of Wieferich; the conjecture remained not shown until today, although J.H. Silverman was able to show in 1988 that if the Conjecture ABC is valid, then for all Integer positive has > 1, it exists an infinity of prime numbers p such as $p^2 \,$ does not divide not $a^ {p-1} - 1 \,$ .

Properties of the prime numbers of Wieferich

It can be shown that a factor first p of a Nombre of Mersenne $M_q = 2^q - 1 \,$ is a prime number of Wieferich if $p^2 \,$ divides $2^q - 1 \,$ ; from this, it follows immediately that a Prime number of Mersenne cannot be a prime number of Wieferich. Also, if p is a prime number of Wieferich, then $2^ {p^2} \ equiv 2 \,$ MOD $p^2 \,$ .

Prime numbers of Wieferich and the last theorem of Fermat

The following theorem connecting the prime numbers of Wieferich and the Dernier theorem of Fermat was proven by Wieferich in 1909:

Is p a prime number, and are X , there , Z of the natural numbers such as $x^p + y^p + z^p = 0 \,$ . Moreover, let us suppose that p does not divide the produced xyz . Then p is a prime number of Wieferich.

In 1910, Mirimanoff was able to develop the theorem by showing that, if the prérequis theorem remains valid for a certain prime number p , then p must also divide $3^ {p-1} \,$ . The prime numbers of this kind were called the prime numbers of Mirimanoff on the occasion, but the name was not general mathematical use.

See too

Prime number of Wilson
Prime number of Wall-Sun-Sun
Prime number of Wolstenholme

External bond

the glossary of the prime numbers: prime numbers of Wieferich (in English)
MathWorld: Prime number of Wieferich (in English)
State of the research of the prime numbers of Wieferich (in English)

Readings to go further

A. Wieferich, " Zum letzten Fermat'schen Theorem ", Newspaper für Queen Angewandte Maths., 136 (1909) 293-302
NR. G.W.H. Beeger, " One has new box off the congruence 2^{p − 1} = 1 (p²) , Messenger off Maths, 51 (1922), 149-150
W. Meissner, " Über die Teilbarkeit von 2p^p − 2 durch das Quadrat der Primzahl p=1093 , Sitzungsber. Akad. D. Wiss. Berlin (1913), 663-667
J.H. Silverman, " Wieferich' S criterion and the ABC-conjecture " , Newspaper off Number Theory, 30:2 (1988) 226-237

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