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moneybird Group Title

Perfect power question

  • 2 years ago
  • 2 years ago

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  1. moneybird Group Title
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    For what primes p is \[3^p + 4^p\] a perfect power?

    • 2 years ago
  2. asnaseer Group Title
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    for p=2 we get:\[3^2+4^2=5^2\]

    • 2 years ago
  3. moneybird Group Title
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    yeah 2 is the only even prime too. so you can factor 3^p + 4^p to (3+4) (3^(p-1) - 4 * 3^(p-2) ... + 4^(p-1))

    • 2 years ago
  4. Stom Group Title
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    but can there be other such primes too?

    • 2 years ago
  5. moneybird Group Title
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    Other than p = 2, 3^p + 4^p must be divisible by 7 i think

    • 2 years ago
  6. mukushla Group Title
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    so we lookin for the solutions of this equation\[3^p+4^p=q^n\]p is prime and q,n>1

    • 2 years ago
  7. mukushla Group Title
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    for p>7 \[3^p+4^p\]is divisible by \(7\) but not by \(7^2\) since\[3^{p-1}-4\times3^{p-2}+4^2\times3^{p-3}-...-3\times4^{p-2}+4^{p-1} \ \ \equiv p3^{p-1} \ \ \text{mod} \ 7\]so there is no solution for p>7 checking for p=2,3,7 gives only solution (p,q,n)=(2,5,2)

    • 2 years ago
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