2015 USAMO Problems/Problem 5

Look at the statement $a^4+b^4=e^5$ . This can be viewed as a special case of Beal's Conjecture (http://en.wikipedia.org/wiki/Beal%27s_conjecture), stating that the equation $A^x+B^y=C^z$ has no solutions over positive integers for $gcd(a, b, c) = 1$ and $x, y, z > 2$ . This special case was proven in 2009 by Michael Bennet, Jordan Ellenberg, and Nathan Ng, as $(x, y, z) = (2, 4, n)$ . This case $a^4+b^4=e^5$ is obviously contained under that special case, so $a$ and $b$ must have a common factor greater than $1$ .

Call the greatest common factor of $a$ and $b$ $f$ . Then $a = f \cdot a_1$ for some $a_1$ and likewise $b = f \cdot b_1$ for some $b_1$ . Then consider the quantity $ac+bd$ .

$ac+bd = f \cdot a_1 \cdot c + f \cdot b_1 \cdot d = f\cdot(a_1 \cdot c + b_1 \cdot d)$ .

Because $c$ and $d$ are both positive, $(a_1 \cdot c + b_1 \cdot d) > 1$ , and by definition $f > 1$ , so $ac+bd$ is composite.

~BealsConjecture

Solution 2

A more conventional approach, using proof by contradiction:

Without loss of generality, assume $a>d$

Since $a^4 +b^4=c^4+d^4$ , It is obvious that $b<c$

We construct the equation $(a^4 +b^4)c^2d^2-(c^4+d^4)a^2b^2=(a^2c^2-b^2d^2)(a^2d^2-b^2c^2)$ (the right side is the factorization of the left) Which factors into: $e^5(cd-ab)(cd+ab)=(ac-bd)(ac+bd)(ad-bc)(ad+bc)$ (by using $a^4 +b^4=c^4+d^4=e^5$ and factoring)

If $ac-bd$ or $ad-bc$ equal zero, then $a:b$ equals $c:d$ or $d:c$ respectively, which is impossible because $a^4 +b^4=c^4+d^4$ and because $a,b,c,d,e$ are distinct. Therefore the right side of the equation above is non-zero and the left side must be divisible by $ac+bd$

If $ac+bd$ were prime,

We have $ac+bd-(cd+ab)=(a-d)(c-b)>0$

Which means that $ac+bd>cd+ab>cd-ab$ and neither $cd+ab$ nor $cd-ab$ can be multiples of $ac+bd$ meaning $e$ must be a multiple of $ac+bd$ and $e=k(ac+bd)$ for some integer $k$