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 March 6th, 2012, 05:36 PM #1 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Fun Diophantine Equations Prove that there are no integer solutions to either of the following equations: $2a^6 + 2b^6 + 2c^6= d^6$ $2a^8 + 4b^8 + 8c^8= d^8$
 March 14th, 2012, 02:52 PM #2 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Re: Fun Diophantine Equations Solution to the first equation: $2a^6 + 2b^6 + 2c^6= d^6$ has no solutions in integers. Consider the equation mod 7. All of $a^6, b^6, c^6, d^6$ must be congruent to 0 or 1 (Fermat's Little Theorem tells us this). There are obviously no solutions such that any of $a^6, b^6, c^6$ is congruent to 1 since then the left side would be congruent to either 2, 4, or 6, and the right side can only be 0 or 1. Hence the only possibility is that all of $a^6, b^6, c^6, d^6$ are congruent to 0, and therefore $a, b, c, d$ are all divisible by 7. But if such a solution exists, then $2(a/7)^6 + 2(b/7)^6 + 2(c/7)^6= (d/7)^6$ is also a solution. If all of the numbers $a/7, b/7, c/7, d/7$ are not divisible by 0, we are done. But if they are all divisible by 0, we can continue to take out factors of 7. Since eventually this process must stop (each number can have only finitely many factors of 7), we arrive at the contradiction that there is a solution not of the form $a, b, c, d= 0$ mod 7 and the proof is complete. It is important to note that we could not use this method to show that $2a^6 + 2b^6 + 2c^6 + 2d^6= e^6$ has no solutions in integers, because the equation holds mod 7 if each of the numbers is congruent to 1, since 8 = 1 (mod 7). It is also essential that each of the numbers on the left side is being multiplied by a factor of at least 2. Consider the equation $a^6 + 2b^6 + 2c^6= d^6$. If $a^6, d^6$ are both congruent to 1 and the others to 0, then the equation holds. In general, we may state that the equation $\sum_{i=1}^k a_ix_i^{p-1} = x_{k+1}^{p-1}$ has no solutions in integers if p is prime, each of the coefficients $a_i$ is at least 2, and $\sum_{i=1}^k a_i < p=$, to be proven by the above method.

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