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estudier
 3 years ago
2^k 15 is square. Find all k (integers)
estudier
 3 years ago
2^k 15 is square. Find all k (integers)

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klimenkov
 3 years ago
Best ResponseYou've already chosen the best response.0Does that mean \(2^k15=a^2, a\in \mathbb N\)?

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1Looks like a square in integers to me:)

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1OK, naturals if u insist

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1Number theory type question.....

estudier
 3 years ago
Best ResponseYou've already chosen the best response.13215 = 17 is not square

hartnn
 3 years ago
Best ResponseYou've already chosen the best response.0next one seems a bit far

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1If u are saying there are no more solutions, saying it is a bit far is insufficient evidence..:)

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1Hint Consider odd and even k separately

shubhamsrg
 3 years ago
Best ResponseYou've already chosen the best response.1maybe i got somewhere 2^k = m^2 or 2m^2 for 2^k = m^2 m^2  15 = k^2 (m+k)(mk) = 1*3*5 on comparison, m+k = 5 and mk =3 =>m=4 >1st solution again on comparison, m+k =15 mk =1 m=8 >2nd solution just have to see through 2m^2 now..

shubhamsrg
 3 years ago
Best ResponseYou've already chosen the best response.1the k which i used on RHS is different from 2^k

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1@shubhamsrg That looks nearly right for even k....

ganeshie8
 3 years ago
Best ResponseYou've already chosen the best response.42^k15 = p^2 2^kp^2 = 15 lets say k is even, k = 2t 2^2t  p^2 = 15 (2^t + p)(2^tp) = 15 solving, we get few solutions lets say k is off, k = 2t+1 2^2t+1  p^2 = 15 (2^2t+1/2 + p)(2^2t+1/2  p) = 15 solving we may not get any solutions on cursory check.. only k = 4,6 are the solutions

shubhamsrg
 3 years ago
Best ResponseYou've already chosen the best response.1similar to what i was going to write..

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1For the odd case, use the fact that any square divided by 3 leaves a remainder of 0 or 1

ganeshie8
 3 years ago
Best ResponseYou've already chosen the best response.4for the even case, k = {4,6} for the odd case : 2.2^2t+1/2 = 16 2.2^2t+1/2 = 8 we need to solve above both

ganeshie8
 3 years ago
Best ResponseYou've already chosen the best response.4again we get the same solutions for odd case k = {4, 6}

estudier
 3 years ago
Best ResponseYou've already chosen the best response.1K, for the odd you can also say that 2^k has remainder 2 on division by 3 and then so does 2^k 15 which is not 0 or 1 so there are no solutions when k is odd
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