Brocard and Ramanujan's problem

The Brocard's problem is an unsolved problem from the mathematical field of number theory . It deals with the question of which faculties are predecessors of square numbers, i.e. which solutions of ${\ displaystyle (m; n)}$

{\ displaystyle n! + 1 = m ^ {2}}

there are in natural numbers .

The only known solutions are ${\ displaystyle (m; n)}$

{\ displaystyle (5; 4), (11; 5)}

and .

{\ displaystyle (71; 7)}

Numerical calculations show that there are no other solutions for . ${\ displaystyle n <10 ^ {9}}$

It is named after Henri Brocard and Srinivasa Ramanujan .

literature

Bruce Berndt , William Galway: The Brocard – Ramanujan diophantine equation n! + 1 = m ² . In: The Ramanujan Journal , Volume 4, 2000, pp. 41-42, math.uiuc.edu (PDF).

Web links

Eric W. Weisstein : Brocard's Problem . In: MathWorld (English).
Eric W. Weisstein : Brown Numbers . In: MathWorld (English).
Ed Copeland: Brown Numbers . In: Numberphile . Brady Haran. Retrieved January 16, 2015.