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Find all prime numbers $p$, for which $$2x^{p-1} + 2009= y ^{p-1}$$ has an infinte number of solutions in non-negative integers? This is a contest problem. Any ideas?

EDIT: Here is the progress i made. Help would be welcomed. Let's rewrite the equation as
$$2x^{p-1} + 2009-y ^{p-1}=0$$
From Fermat's theorem we have that

$0\equiv 2x^{p-1} + 2009-y ^{p-1}\equiv 2008,\ 2009$ ,$2010$ or $2011\pmod p$
So obviously $p$ can only be a prime divisor of $2008,\ 2009$ ,$2010$ or $2011$, $=>p=2, 3, 5, 7, 41,67, 251$ or $2011$

  • $p=2$ we get $2x+2009-y=0$ which has infinite number of solutions.
  • $p=5\ => 2x^4+2009=y^4$. $y^4\equiv {0,1,15} \pmod {16},\ 2x^4+2009\equiv{8,9,10} \pmod {16} =>$ no solutions.

  • $p=7\ => 2x^6+7.7.41=y^6$ $=> 2x^6-y^6\equiv 0\pmod 7$.
    From Fermat's theorem $2x^6\equiv 0$ or $2 \pmod 7$, $y^6 \equiv 1$ or $0 \pmod 7$ =>$x=7k\ ,\ y=7l $ => $2009\equiv 0 \pmod {7^6}$ which is false => no possible solutions for $p=7$.

  • The case for $p=41$ is solved analogously to $p=7$.

I have no ideas for $p=3, 5 ,67, 251$ or $ 2011$ yet.

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  • $\begingroup$ If $x$ and $p$ are coprime, you can apply Fermat theorem, if not, $x^{p-1}\equiv0\pmod p$. The same thing with $y$. Give it a try. Try also saying that $gdc(x,y)=d$, which yeilds $\exists u,v:\; xu+yv=d$. $\endgroup$ Aug 5 '16 at 15:18
  • $\begingroup$ Did you find any prime numbers $p$ for which the relation admits an infinite number of solutions $x,y$? Generally you are expected to supply more context than a bare problem statement. Why is the problem interesting or difficult? What do you already know about approaches to this problem? $\endgroup$
    – hardmath
    Aug 5 '16 at 22:52
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The primes are $p=2$ and $p=3.$ Any larger exponent forces a finite number of solutions, Thue's Theorem. There are infinitely many solutions to $x^2 - 2 y^2 = 2009,$ every sixth pair related by $$ x_{n+2} = 6 x_{n+1} - x_n, $$ $$ y_{n+2} = 6 y_{n+1} - y_n. $$ That is, there are six orbits of solutions under the action of the (oriented) automorphism group of the quadratic form.

as in $$ (47,10); (181,124); (1039, 734); (6053, 4280);... $$ and five others.

jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
  Automorphism matrix:  
    3   4
    2   3
  Automorphism backwards:  
    3   -4
    -2   3

  3^2 - 2 2^2 = 1

 x^2 - 2 y^2 = 2009

Fri Aug  5 13:57:33 PDT 2016

x:  47  y:  10 ratio: 4.7  SEED   KEEP +- 
x:  49  y:  14 ratio: 3.5  SEED   KEEP +- 
x:  53  y:  20 ratio: 2.65  SEED   KEEP +- 
x:  79  y:  46 ratio: 1.71739  SEED   BACK ONE STEP  53 ,  -20
x:  91  y:  56 ratio: 1.625  SEED   BACK ONE STEP  49 ,  -14
x:  101  y:  64 ratio: 1.57812  SEED   BACK ONE STEP  47 ,  -10
x:  181  y:  124 ratio: 1.45968
x:  203  y:  140 ratio: 1.45
x:  239  y:  166 ratio: 1.43976
x:  421  y:  296 ratio: 1.4223
x:  497  y:  350 ratio: 1.42
x:  559  y:  394 ratio: 1.41878
x:  1039  y:  734 ratio: 1.41553
x:  1169  y:  826 ratio: 1.41525
x:  1381  y:  976 ratio: 1.41496
x:  2447  y:  1730 ratio: 1.41445
x:  2891  y:  2044 ratio: 1.41438
x:  3253  y:  2300 ratio: 1.41435
x:  6053  y:  4280 ratio: 1.41425
x:  6811  y:  4816 ratio: 1.41424
x:  8047  y:  5690 ratio: 1.41424
x:  14261  y:  10084 ratio: 1.41422
x:  16849  y:  11914 ratio: 1.41422
x:  18959  y:  13406 ratio: 1.41422
x:  35279  y:  24946 ratio: 1.41421
x:  39697  y:  28070 ratio: 1.41421
x:  46901  y:  33164 ratio: 1.41421
x:  83119  y:  58774 ratio: 1.41421
x:  98203  y:  69440 ratio: 1.41421
x:  110501  y:  78136 ratio: 1.41421
x:  205621  y:  145396 ratio: 1.41421
x:  231371  y:  163604 ratio: 1.41421
x:  273359  y:  193294 ratio: 1.41421
x:  484453  y:  342560 ratio: 1.41421
x:  572369  y:  404726 ratio: 1.41421
x:  644047  y:  455410 ratio: 1.41421
x:  1198447  y:  847430 ratio: 1.41421
x:  1348529  y:  953554 ratio: 1.41421
x:  1593253  y:  1126600 ratio: 1.41421
x:  2823599  y:  1996586 ratio: 1.41421
x:  3336011  y:  2358916 ratio: 1.41421
x:  3753781  y:  2654324 ratio: 1.41421
x:  6985061  y:  4939184 ratio: 1.41421
x:  7859803  y:  5557720 ratio: 1.41421
x:  9286159  y:  6566306 ratio: 1.41421
x:  16457141  y:  11636956 ratio: 1.41421
x:  19443697  y:  13748770 ratio: 1.41421

Fri Aug  5 13:58:13 PDT 2016

 x^2 - 2 y^2 = 2009

jagy@phobeusjunior:~$ 
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