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
![p> 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/0502012bc3b4e73e6f3c2f4748feaab3fd3c350d)
-
.
Examples
![{1 ^ {{2}} + 2 ^ {{2}}} = 5 \ equiv {5} = {3+ (3-1)!} {\ Pmod {9}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04150f9e9691f80ac56c0388f60c431ac72f66c)
![{1 ^ {{4}} + 2 ^ {{4}} + 3 ^ {{4}} + 4 ^ {{4}}} = 354 \ equiv {4} \ equiv {29} = {5+ ( 5-1)!} {\ Pmod {25}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c80489087c1486dd226ff5103a60762209c3faf)
![{1 ^ {{6}} + 2 ^ {{6}} + 3 ^ {{6}} + 4 ^ {{6}} + 5 ^ {{6}} + 6 ^ {{6}}} = 67.171 \ equiv {41} \ equiv {727} = {7+ (7-1)!} {\ Pmod {49}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70a11add2eb2e1e293cc28d3b2dfb7cbd8c1851f)
![{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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/453c1f35beba7831ea67ff57a787bc1df69a39f5)
Derivation of the formula according to Sierpiński
According to Wilson's theorem is the quotient
![r_ {0}: = {\ frac {(p-1)! + 1} {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a0dfb7985b8567f3c235ceb7efbd51b37db7b7)
an integer .
According to Fermat's little theorem, the quotients are in the same way
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For
also whole numbers.
From this it follows first
-
For
such as
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.
This results on the one hand
-
and then
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,
On the other hand , the binomial theorem applies
-
and thus
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.
Taken together , you have the congruence
-
.
If one goes into the equation with this congruence
-
,
so it finally results
-
.
literature
References and comments
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^ 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 ).
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^ Siegfried Gottwald (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch, Thun / Frankturt / Main 1990, ISBN 3-8171-1164-9 , p. 283 .
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↑ Here it goes into the fact that when multiplying - terms from two or more brackets, the product modulo has the value zero.
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↑ At this point it comes into play that and thus as a prime number is necessarily odd.