Erdős-Selfridge's theorem

The set of Erdos Selfridge (not to be confused with a same set of game theory) is a theorem of number theory , a partial area of mathematics . It goes back to the two mathematicians Paul Erdős and John L. Selfridge and deals with a classic problem about Diophantine equations .

This problem relates to the question of whether a product of several consecutive natural numbers can be a true power of natural numbers. With their sentence Erdős and Selfridge provide a complete solution to this problem and give a negative answer to the question.

formulation

The sentence is:

If two or more consecutive natural numbers are multiplied with one another , the product is not a real power with natural base and exponent.

Or equivalent and a little more formal:

The Diophantine equation

${\displaystyle (n+1)(n+2)\cdots (n+k)=m^{r}}$

is unsolvable for ( integer ) . ${\displaystyle k\geq 2,r\geq 2,m\geq 2}$

Related problems

Paul Erdős has also solved two related problems, which are based on the same question as in the above sentence, also in the negative sense. The two results are as follows:

1. The product of two or more consecutive odd natural numbers is not a real power with natural prime and exponent. (Erdős 1939).
2. The binomial coefficient is for natural numbers with no real power with natural base and exponent. ${\displaystyle {\binom {n}{k}}}$ ${\displaystyle n,k}$ ${\displaystyle k\geq 4,n\geq 2k}$(Erdős 1951).

literature

• Paul Erdős, JL Selfridge : The product of consecutive integers is never a power . In: Illinois J. Math . tape 19 , 1975, p. 292-301 ( projecteuclid.org [PDF] MR0376517 ).
• Paul Erdős, János Surányi : Topics in the Theory of Numbers . Translated from Hungarian by Barry Guiduli (=  Undergraduate Texts in Mathematics ). 2nd Edition. Springer Verlag, New York ( inter alia ) 2003, ISBN 0-387-95320-5 ( MR1950084 ).
• 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 ).

References and comments

1. ↑ ^{a b} Erdős, Surányi: pp. 226–227.
2. ^ ^{A b c} 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. 86-87 ( MR0930670 ).
3. ↑ Erdős, Selfridge: The product of consecutive integers is never a power. In: Illinois J. Math . tape 19 , p. 292 ff .