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Power 2 Sum Power 3 And 5 (Posted on 2008-10-04) Difficulty: 4 of 5
Determine all possible pair(s) of nonnegative integers (P, Q), that satisfy this equation:

†††††††††††††††††††††††††††††††††††††††††2P = 3Q + 5

  Submitted by K Sengupta    
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Solution: (Hide)
Substituting P = 0,1,2,3,4,5 in turn in the given equation, it follows that: (P, Q) = (3,1), (5,3) are the only solutions whenever P ≤ 5.

If possible, we suppose that valid solutions exist for P > 5. In that situation, 3^Q + 5 is divisible by 2^6 = 64.
On the other hand, 3^16 = 1 mod 64, and we observe that: 3^11 = - 5 mod 64, and 3 ^k ≠ -5 mod 64 for other values of k from 0 to 15.

Accordingly, Q must be of the form: Q = 16k + 11.

Now, by Fermatís Little Theorem, we have:
3^16 = 1 mod 17, so that:
3 ^(16k+11) = 3^11 = 7 mod 17.
Accordingly, 2^P = 3^11 + 5 = 12 mod 17.

However, the valid residues of 2^P, in mod 17 are 2, 4, 8, 16, 15, 13, 9, 1, and therefore we cannot have: 2^P = 12 mod 17. This contradicts our supposition that there are valid solutions whenever P > 5.

Consequently, (P, Q) = (3,1), (5,3) are the only possible nonnegative solutions to the given problem.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(4): almost proof NOT TRUESteve Herman2008-10-05 21:24:32
re(3): Charlie --asking moreCharlie2008-10-05 19:12:03
Questionre(2): Charlie --asking moreAdy TZIDON2008-10-05 18:18:11
re(3): almost proof NOT TRUEAdy TZIDON2008-10-05 18:08:55
re: asking Charlie -- more detailCharlie2008-10-05 12:51:05
re: asking CharlieCharlie2008-10-05 11:29:30
Some Thoughtsre(2): almost proof NOT TRUESteve Herman2008-10-05 10:58:05
Some Thoughtsre: almost proof NOT TRUEAdy TZIDON2008-10-05 05:24:35
almost proof ;-)Daniel2008-10-05 00:46:33
Questionasking CharlieAdy TZIDON2008-10-04 20:12:39
Solutioncomputer solution--no proof of completenessCharlie2008-10-04 15:48:05
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