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Well, the question is so:

Suppose P is the set of all primes, p, that satisfy the condition that $8^p+15^p$ is a perfect square.

Find the sum of the elements of P.

Now, over here, I found out a few bits of information. However, I've no idea how to use them, and whether they can even be used.

For example, it should be obvious that 2 belongs to P. As $64+225 = 289 = 17^2$ Also,

$15\equiv -8 \pmod{23}$ So, for all primes greater than 2, (which are obviously odd)

$8^p+ 15^p\equiv 8^p+ (-8)^p\equiv 8^p-8^p \equiv 0 \pmod{23}$

Suppose $8^p+15^p = x^2$, then, $x^2 \equiv 0 \pmod{23}$,

And, as 23 is a prime, this means that, $x \equiv 0 \pmod{23}$ and $x^2 \equiv 0 \pmod{23^2}$.

I also found out that,

$8^p +15^p \equiv 8^{p-(p-1)}+15^{p-(p-1)} \pmod{p}$, by Euler's Theorem.

Therefore,

$x^2\equiv 8^p+15^p \equiv 8+15 \equiv 23 \pmod{p}$.

Hmm. I'd really appreciate it if, instead of giving me a solution flat out, someone could point me in the right direction and perhaps offer a slight push.

Thanks a lot, people.

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3 Answers

Consider what perfect squares are modulo 5 (if you don't know that, work out the possibilities) and then consider the equation $8^p + 15^p$ mod 5. $15^p$ should be easy, and what possibilities are there for $8^p$?

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Mod 3 is even easier. –  EuYu Nov 6 '12 at 18:25
    
Hey! Thanks a lot for that! I already did the 8^p+15^p mod 5 and mod 3 earlier. I even did mod 10. What was really great was looking at the squares mod 5 and mod 3. Fabulous patterns begin to emerge, even with other modular bases. I'd never have suspected that. I got my answer. 2 is the only prime that works, because the modulars equate only if p is even, which is only possible when p = 2. I even came across excerpts from Gauss' Disquisitiones on similar patterns. Beautiful. –  MWarsi Nov 7 '12 at 9:08
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Let $v_p(n)$ denote the maximum prime power in $n$. That is if, $v_p(n)=\alpha\implies p^{\alpha}\mid n$ but $p^{\alpha+1}\nmid n$.

As, $23\mid 8+15$

By Lifting the Exponent Lemma:

$v_{23}(8^p+15^p)=v_{23}(8+15)+v_{23}(p)=1+v_{23}(p)$.

For, odd prime $p$, $23\mid 8^p+15^p$ but $23^2\mid 8^p+15^p\iff p=23$.

So, answer $=2+23=25$.

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This cannot possibly be right $$8^p + 15^p \equiv (-1)^p \pmod3$$ Squares are $0$ or $1$ mod $3$. This necessarily means $2\mid p$ –  EuYu Nov 6 '12 at 18:45
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Oops! Yes I missed that. –  Grobber Nov 7 '12 at 16:33
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Note that for any $n$, we have $n^2\equiv 0\:(mod\: p)$ or $n^2\equiv 0\:(mod\: p)$. (To see it, check for each of the cases $n=3k$, $n=3k+1$, $n=3k-1$.)

Since $3|15$, then $3|15^m$ for all $m>0$, so $3|15^p$ for all (positive) primes $p$, and so $15^p\equiv 0\:(mod\: p)$ for any prime $p\in P$. On the other hand, $3$ fails to divide any power of $8$, so $8^p\not\equiv 0\:(mod\: p)$ for any prime $p\in P$.

With those observations (and the fact that $2\in P$) in mind, we'll be able to determine precisely what the set $P$ is.

Assume, then that $p\in P$, so that $8^p+15^p=n^2$ for some $n$. We cannot have $n^2=0\:(mod\: 3)$, for then $8^p\equiv 8^p+15^p\equiv 0\:(mod\:p)$, which we already determined to be false. Thus, we must have $n^2\equiv 1\:(mod\: p),$ and so $$8^p\equiv 8^p+15^p\equiv 1\:(mod\: p).\tag{1}$$ Writing $8=9-1$, binomial expansion gives us $$8^p=\sum_{k=0}^p\binom{p}{k}(-1)^k9^{p-k}=(-1)^p+3\sum_{k=0}^{p-1}\binom{p}{k}(-1)^k3^{2p-2k-1},$$ so $8^p\equiv(-1)^p\: (mod\: p),$ and combining this with $(1)$ gives us $$(-1)^p\equiv 1\:(mod\: 3).\tag{2}$$ Since $-1\not\equiv 1\:(mod\: 3)$, then $p$ cannot be odd.

Hence, $P=\{2\}$.

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