# Quasi-Fermat Equasion

**URL:** <https://boards.straightdope.com/t/quasi-fermat-equasion/617171>\
**Category:** Factual Questions\
**Created:** [March 29, 2012, 4:32pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171 "2012-03-29T16:32:06Z")\
**Posts on this page:** 10\
**Page:** 1

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**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\
**Post date:** [March 29, 2012, 4:32pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/1 "2012-03-29T16:32:06Z")

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Is there a solution to X^e + Y^e = Z^e where X,Y, and Z are all whole numbers?

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**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\
**Post date:** [March 29, 2012, 4:38pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/2 "2012-03-29T16:38:27Z")

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X = 0, Y = Z works, but that’s trivial.

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**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\
**Post date:** [March 29, 2012, 4:38pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/3 "2012-03-29T16:38:44Z")

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Just to be clear, you’re asking about e, the constant 2.718… ?

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**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\
**Post date:** [March 29, 2012, 4:59pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/4 "2012-03-29T16:59:07Z")

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Yes, The base of the natural log.

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**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\
**Post date:** [March 29, 2012, 5:03pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/5 "2012-03-29T17:03:08Z")

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More generally, is it even known if there _any_ irrational exponents with solutions? (certain fractional exponents do work) What made you wonder about ‘e’ in particular?

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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:** [March 29, 2012, 5:32pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/6 "2012-03-29T17:32:47Z")

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One usually encounters e in the bottom of an exponentiation, not the top. Seeing something raised to the e power is is already a sign that things are going to get wonky.

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**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\
**Post date:** [March 29, 2012, 5:34pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/7 "2012-03-29T17:34:50Z")

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Well, e is the magical constant 😉 when involved in logs and exponents.

Check out this graphical calculator:

> **[Desmos | Graphing Calculator](https://www.desmos.com/calculator)**
>
> Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

Inputting y=x^1/x displays y max when x=e

X^n + Y^n = Z^n

n=1…simple arithmetic  
n=2…Fermat’s Last Theorem (3:4:5, 5:12:13, etc)

n will not work for whole numbers greater than 2.

e is not a whole number…and neat mathematical stuff happens when you involve e.

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**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\
**Post date:** [March 29, 2012, 6:23pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/8 "2012-03-29T18:23:55Z")

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> [@KarlGauss](#):
>
> More generally, is it even known if there _any_ irrational exponents with solutions? (certain fractional exponents do work)

[Here is a page](http://math.williams.edu/fermats-last-theorem-for-fractional-exponents/)that indicates that no solution exists for rational exponents greater than 2, but that real values solutions definitely exist. That said I would be very surprised if there existed solutions for e.

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**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\
**Post date:** [March 29, 2012, 8:47pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/9 "2012-03-29T20:47:45Z")

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> [@Buck\_Godot](#):
>
> [Here is a page](http://math.williams.edu/fermats-last-theorem-for-fractional-exponents/)that indicates that no solution exists for rational exponents greater than 2, but that real values solutions definitely exist. That said I would be very surprised if there existed solutions for e.

I am more than impressed that you found that. Very nice.

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [March 30, 2012, 11:30pm UTC](https://boards.straightdope.com/t/quasi-fermat-equasion/617171/10 "2012-03-30T23:30:28Z")

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This does not answer the OP (I very much doubt that any knows the answer to that one), but here is one thing to think about. Since 6^3 + 7^3 \> 8^3 and 6^4 + 7^4 \< 8^4, the function f(x) = 6^x + 7^x - 8^x is positive when x = 3 and negative when x = 4 and therefore 0 at some point in the interval 3 \< x \< 4. Almost surely irrational and very likely transcendental.

BTW, the equation x^2 + y^2 = z^2 is not Fermat’s equation. It was known to Euclid to have infinitely many solutions. He knew that if you chose numbers m \> n with no common divisor and one even, then x = 2mn, y = m^2 - n^2 and z = m^2 + n^2 you always got a “primitive” solution (one in which x, y and z had no common divisor). It is unclear if he realized that this gives all the primitive solutions (but it does). The Fermat equation is x^n + y^n = z^n for n \> 2.