Lerch's theorem (number theory)

The set of Lerch is a theorem of elementary number theory , one of the part areas of mathematics . It goes back to the Austro-Czech mathematician Matyáš Lerch and contains a formula about the congruence of certain power sums for odd prime numbers . The formula is also known as Lerch's formula in elementary number theory . Their derivation is based on Wilson's theorem and Fermat's little theorem .

The formula

Lerch's formula says:

Every prime number     fulfills the congruence ${\displaystyle p>2}$

${\displaystyle {1^{p-1}+2^{p-1}+\cdots +(p-1)^{p-1}}\equiv {p+(p-1)!}{\pmod {p^{2}}}}$  .

Examples

${\displaystyle {1^{2}+2^{2}}=5\equiv {5}={3+(3-1)!}{\pmod {9}}}$

${\displaystyle {1^{4}+2^{4}+3^{4}+4^{4}}=354\equiv {4}\equiv {29}={5+(5-1)!}{\pmod {25}}}$

${\displaystyle {1^{6}+2^{6}+3^{6}+4^{6}+5^{6}+6^{6}}=67.171\equiv {41}\equiv {727}={7+(7-1)!}{\pmod {49}}}$

${\displaystyle {1^{10}+2^{10}+3^{10}+4^{10}+5^{10}+6^{10}+7^{10}+8^{10}+9^{10}+10^{10}}=14.914.341.925\equiv {21}\equiv {3.628.811}={11+(11-1)!}{\pmod {121}}}$

Derivation of the formula according to Sierpiński

According to Wilson's theorem is the quotient

${\displaystyle r_{0}:={\frac {(p-1)!+1}{p}}}$

an integer .

According to Fermat's little theorem, the quotients are in the same way

${\displaystyle r_{k}:={\frac {k^{p-1}-1}{p}}}$   For   ${\displaystyle k=1,\ldots ,p-1}$

also whole numbers.

From this it follows first

${\displaystyle k^{p-1}={pr_{k}+1}}$   For   ${\displaystyle k=1,\ldots ,p-1}$

such as

${\displaystyle (p-1)!={pr_{0}-1}}$  .

This results on the one hand

${\displaystyle {\begin{aligned}{((p-1)!)}^{p-1}&={1^{p-1}\cdot 2^{p-1}\cdot \;\cdots \;\cdot (p-1)^{p-1}}\\&=(pr_{1}+1)\cdot (pr_{2}+1)\cdot \;\cdots \;\cdot (pr_{p-1}+1)\\\end{aligned}}}$  

and then

${\displaystyle {\begin{aligned}{((p-1)!)}^{p-1}&{\equiv }\;\;pr_{1}\cdot 1^{p-2}+pr_{2}\cdot 1^{p-2}+\cdots +pr_{p-1}\cdot 1^{p-2}+1^{p-1}{\pmod {p^{2}}}\\&{\equiv }\;\;p\cdot (r_{1}+r_{2}+\cdots +r_{p-1})+1{\pmod {p^{2}}}\\\end{aligned}}}$  ,

On the other hand , the binomial theorem applies

${\displaystyle {\begin{aligned}{((p-1)!)}^{p-1}&=(pr_{0}-1)^{p-1}\\&=\sum _{j=0}^{p-1}{\binom {p-1}{j}}\cdot (pr_{0})^{j}\cdot (-1)^{p-1-j}\\\end{aligned}}}$  

and thus

${\displaystyle {\begin{aligned}{((p-1)!)}^{p-1}&{\equiv }\;\;1-(p-1)\cdot pr_{0}{\pmod {p^{2}}}\\&{\equiv }\;\;1-p^{2}r_{0}+pr_{0}{\pmod {p^{2}}}\\&{\equiv }\;\;1+pr_{0}{\pmod {p^{2}}}\\\end{aligned}}}$  .

Taken together , you have the congruence

${\displaystyle p\cdot (r_{1}+r_{2}+\cdots +r_{p-1})\equiv \;\;pr_{0}{\pmod {p^{2}}}}$  .

If one goes into the equation with this congruence

${\displaystyle {\begin{aligned}1^{p-1}+2^{p-1}+\cdots +(p-1)^{p-1}&=(pr_{1}+1)+(pr_{2}+1)+\;\cdots \;+(pr_{p-1}+1)\\&=p\cdot (r_{1}+r_{2}+\;\cdots \;+r_{p-1})+p-1\\\end{aligned}}}$  ,

so it finally results

${\displaystyle {\begin{aligned}1^{p-1}+2^{p-1}+\cdots +(p-1)^{p-1}&{\equiv }\;\;pr_{0}+p-1{\pmod {p^{2}}}\\&{\equiv }\;\;p+pr_{0}-1{\pmod {p^{2}}}\\&{\equiv }\;\;p+(p-1)!{\pmod {p^{2}}}\\\end{aligned}}}$  .

literature

• Matyáš Lerch : On the theory of Fermat's quotient ${\displaystyle {\frac {a^{p-1}-1}{p}}=q(a)}$ . In: Mathematical Annals . tape 60 , 1905, pp. 471-490 , doi : 10.1007 / BF01561092 ( MR1511321 ).
• 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 ).
• Siegfried Gottwald (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch, Thun / Frankturt / Main 1990, ISBN 3-8171-1164-9 .

References and comments

1. ^ 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 , pp. 225-226 ( MR0930670 ).
2. ^ Siegfried Gottwald (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch, Thun / Frankturt / Main 1990, ISBN 3-8171-1164-9 , p. 283 .
3. ↑ Here it goes into the fact that when multiplying - terms from two or more brackets, the product modulo has the value zero. ${\displaystyle p}$ ${\displaystyle p^{2}}$
4. ↑ At this point it comes into play that and thus as a prime number is necessarily odd. ${\displaystyle p>2}$