Difference between revisions of "Brocard's problem"

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Are the known solutions, and it was a conjecture of Paul Erdös, that these are the only solutions.
 
Are the known solutions, and it was a conjecture of Paul Erdös, that these are the only solutions.
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==Heuristic arguments==
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- If <math>n>3</math> then <math>n^2<n!</math>  therefore <math>m>n</math>
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- If <math>n>2p</math> LHS is 1 mod <math>p^2</math> so <math>m</math> is likely <math>\pm 1</math> mod <math>p^2</math> making <math>m</math> spread out as <math>n</math> increases.
  
 
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[[Category:Number theory]]

Latest revision as of 19:15, 12 March 2020

Problem Statement

Given $n!+1=m^2$ which pairs $(n,m)$ are the solutions to the given equation.

Known Solutions

\[(n,m)=(4,5),(5,11),(7,71)\]

Are the known solutions, and it was a conjecture of Paul Erdös, that these are the only solutions.

Heuristic arguments

- If $n>3$ then $n^2<n!$ therefore $m>n$

- If $n>2p$ LHS is 1 mod $p^2$ so $m$ is likely $\pm 1$ mod $p^2$ making $m$ spread out as $n$ increases.

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