# Did Cauchy prove Fermat for cardinal numbers?

**URL:** <https://boards.straightdope.com/t/did-cauchy-prove-fermat-for-cardinal-numbers/763766>\
**Category:** Factual Questions\
**Created:** [August 23, 2016, 9:19pm UTC](https://boards.straightdope.com/t/did-cauchy-prove-fermat-for-cardinal-numbers/763766 "2016-08-23T21:19:13Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\
**Post date:** [August 23, 2016, 9:19pm UTC](https://boards.straightdope.com/t/did-cauchy-prove-fermat-for-cardinal-numbers/763766/1 "2016-08-23T21:19:13Z")

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I’ve been re-reading Simon Singh’s book on Fermat and he relates how Cauchy’s attempt to prove Fermat’s Last Theorem was derailed by the issue of unique factorisation via complex numbers. However, Singh leaves it there. Did Cauchy in fact prove Fermat true for cardinal numbers?

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [August 23, 2016, 11:37pm UTC](https://boards.straightdope.com/t/did-cauchy-prove-fermat-for-cardinal-numbers/763766/2 "2016-08-23T23:37:53Z")

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Fermat’s Last Theorem is only stated to apply for cardinal numbers in the first place. If Cauchy had proven it for cardinal numbers, then he would have proven it.

The fact that complex numbers show up in the attempted proof does not mean that anyone was trying to apply the theorem to complex numbers.

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**Author:** ![kk\_fusion](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kk_fusion/32/15276_2.png) [@kk\_fusion](https://boards.straightdope.com/u/kk_fusion)\
**Post date:** [August 24, 2016, 7:34am UTC](https://boards.straightdope.com/t/did-cauchy-prove-fermat-for-cardinal-numbers/763766/3 "2016-08-24T07:34:48Z")

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No, the first rigorous proof for the special case of [regular primes](https://en.wikipedia.org/wiki/Regular_prime) (and their multiples) was published by Ernst Kummer in 1850.

Cauchy and Lamé had only presented an outline of a proof, which they retracted after the flaw concerning irregular primes was pointed out.

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**Author:** ![Quartz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/quartz/32/267_2.png) [@Quartz](https://boards.straightdope.com/u/Quartz)\
**Post date:** [August 24, 2016, 12:40pm UTC](https://boards.straightdope.com/t/did-cauchy-prove-fermat-for-cardinal-numbers/763766/4 "2016-08-24T12:40:35Z")

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Sorry, I meant if you could only use cardinals when factoring.

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [August 24, 2016, 3:39pm UTC](https://boards.straightdope.com/t/did-cauchy-prove-fermat-for-cardinal-numbers/763766/5 "2016-08-24T15:39:26Z")

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But Fermat’s Last Theorem isn’t about factoring. If factoring shows up in a proof (or attempted proof), it’s because the mathematics led there from the original problem statement. And if the mathematics leads there in such a way that irregular factoring is relevant, then irregular factoring is relevant, and you can’t ignore it.