# 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. | ||

{{stub}} | {{stub}} | ||

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+ | [[Category:Number theory]] |

## Latest revision as of 19:15, 12 March 2020

## Problem Statement

Given which pairs are the solutions to the given equation.

## Known Solutions

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

## Heuristic arguments

- If then therefore

- If LHS is 1 mod so is likely mod making spread out as increases.

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