Expert: Paul Klarreich - 2/4/2009

Question

you have positive integers a,b,c,d with a^3=b^5 and c^3 = d^5 and c-a = 32.
Show there is no solution for a,b,c,d as integers

my attempt:
a^3 = b^5 implies that there is a prime factor in a that is a 15th power that
matches a prime factor in b^5 that is also raise to the 15th power or put
another way a = uuuuu and c = vvvvv and c-a = uuuuu-vvvvv = 32 so
uuuuu = 32 + vvvvv. I know the only integral 5th root of 32 is 2 so is it
true that u-v = 2? (I think it is). And by division we get a quotient of
uuuu+uuuv+uuvv+uvvv+vvvv. Now there are a gazillion ways to get a difference
of 2, is there some way to bound this?

Answer
Questioner:   Sombra
Category:  Advanced Math
Private:  No
 
Subject:  rational roots theorem needed?
Question:  
you have positive integers a,b,c,d with a^3=b^5 and c^3 = d^5 and c-a = 32.
Show there is no solution for a,b,c,d as integers

my attempt:
a^3 = b^5 implies that there is a prime factor in a that is a 15th power that
matches a prime factor in b^5 that is also raise to the 15th power or put
another way a = uuuuu and c = vvvvv and c-a = uuuuu-vvvvv = 32 so
uuuuu = 32 + vvvvv. I know the only integral 5th root of 32 is 2 so is it
true that u-v = 2? (I think it is). And by division we get a quotient of
uuuu+uuuv+uuvv+uvvv+vvvv. Now there are a gazillion ways to get a difference
of 2, is there some way to bound this?

.................................
Hi, sombra,

You forgot?  I thought we did this already:

a^3 = b^5 -->  a is a 5th power = x^5, for some x.
c^3 = d^5 -->  c is a 5th power = z^5, for some z.

c - a = 32  --> a + 32 = c

a + 32 = c -->  x^5 + 32 = z^5

--> x^5 + 2^5 = y^5

There was this Frenchman, Zermat, Germat, Hermat, or something -- I can't quite remember the exact name -- who wrote in one of his notebooks, found only after his death:

<In French, of course.>
More generally, x^n + y^n = z^n  has no solutions in positive integers for all n > 2.  I have a marvelous proof of this which the margin is too small to contain.
< end of French>

Well, you and I have now proved he was wrong.  Never mind Wiles and all the others.  All you have to do now, of course, is find the x and the y -- you will be famous.

Aha!  I got it. FERMAT was the name.