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# false counterexamples to Fermat’s last theorem

Like Martin Gardner’s famous 1975 joke that $e^{{\pi\sqrt{163}}}$ is an integer, hoax counterexamples to Fermat’s last theorem typically depend on the loss of machine precision. The following false counterexamples should check out on most scientific calculators.

$\displaystyle 1782^{{12}}+1841^{{12}}$ | $\displaystyle=$ | $\displaystyle 1922^{{12}}$ | ||

$\displaystyle 6107^{6}+8919^{6}$ | $\displaystyle=$ | $\displaystyle 9066^{6}$ | ||

$\displaystyle 3987^{{12}}+4365^{{12}}$ | $\displaystyle=$ | $\displaystyle 4472^{{12}}$ |

Executing the left side of the first equation on a typical scientific calculator and then taking the 12th root of that result will yield 1922. But on software calculators, such as the Mac OS Calculator, the 12th root is given as 1921.99999995495. Rising points out that it is not necessary to carry out any calculations in order to see that the first equation is false: “The left side adds an even number and an odd number; thus that sum must be odd. The right side is even.” In fact, modular arithmetic can be used to show all these equations are false: the left side of the second one is congruent to $5\mod 9$ while the right side is congruent to $3\mod 9$; casting out nines also disproves the third equation.

Taxicab numbers involving $1^{x}$ in one of the expressions, like Ramanujan’s friend 1729, can be used to benchmark machine precision. Any scientific calculator will readily show that $10^{3}+9^{3}\neq 12^{3}$, since $\sqrt[3]{1729}\approx 12.002314$, and for some larger taxicab number a false counterexample will appear. A computer algebra system, on the other hand, can determine the falsehood of a counterexample even if it lacks the machine precision to resolve the numerical difference. For example, on Mathematica, assuming `x, y, z, n`

have been defined: `TrueQ[x^n + y^n == z^n]`

should return `False`

.

Some of these false counterexamples have appeared on episodes of The Simpsons, such as the first one, which appeared in the $\textrm{Homer}^{3}$ segment of “Treehouse of Horror VI,” first aired October 29, 1995, more than a year after Andrew Wiles and his colleagues announced the corrected proof of Fermat’s last theorem.

# References

- 1 Gerald R. Rising, Inside Your Calculator: From Simple Programs to Significant Insights. Hoboken, New Jersey: John Wiley & Sons (2007): Appendix D
- 2 Ray Richmond & Antonia Coffman, The Simpsons: A Complete Guide to Our Favorite Family. New York: HarperCollings (1997): 187

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## Comments

## Another TeX verbatim glitch

I don't know what's wrong with this entry. My top guess is: yet another glitch with verbatim. I'm giving a couple of friends access if they want to take a try at fixing it.

## Re: Another TeX verbatim glitch

BTW, the invalidity of the second "equation"

6107^6+8919^6 = 9066^6

is seen by inferring the last digit of both sides:

LHS is modulo 10 congruent with 7^6+9^6 = 9^3+1^3 = 9+1 = 0,

RHS modulo 10 with 6^6 = 6^3 = 6.

Regards,

Jussi

## Re: Another TeX verbatim glitch

These Fermat "counter examples" demonstrate the power of elementary modular algebra. The first "equation" fails modulo 2, as noted by the author himself. The second one fails module 10, as noted by Pahio. The third one fails modulo 9: both LHS numbers are divisible by 9, the RHS is not.

## Re: Mod arithmetic disproves Fermat counterexamples

> These Fermat "counter examples" demonstrate the power of elementary

> modular algebra. The first "equation" fails modulo 2, as noted by the > author himself. The second one fails module 10, as noted by Pahio. The > third one fails modulo 9: both LHS numbers are divisible by 9, the RHS > is not.

>

"As noted by the author HERself."

Overall, I agree. In fact, this is an opportunity to link to the PM entry that proves that if $a \equiv b \mod c$ then $a^n \equiv b^n \mod c$ for $n > 0$. Only... I can't find that entry... is it missing?

## Re: Mod arithmetic disproves Fermat counterexamples

See 3 in http://planetmath.org/encyclopedia/Congruences.html

## Re: Mod arithmetic disproves Fermat counterexamples

> See 3 in http://planetmath.org/encyclopedia/Congruences.html

Yes, but it wouldn't hurt to say that exponentiation is one such function with integer coefficients. In fact, because exponentiation is not typically expressed in function notation, it is all the more pressing to specifically mention it.