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A quotient and square problem (Posted on 2006-09-23) Difficulty: 4 of 5
Determine all primes m for which the quotient (2m-1 - 1)/m is a perfect square.

  Submitted by K Sengupta    
Rating: 3.0000 (2 votes)
Solution: (Hide)
( We denote < sign, > sign, < = sign and > = sign respectively by lt, gt, lteq and gteq))

If m is a prime which is not equal to 2, then

(2^m -1 ) = (2^(m-1)/2 - 1 )(2^(m-1)/2 + 1 ) is divisible by m in terms of the Fermat's theorem.

So, either (2^(m-1)/2 - 1 )or (2^(m-1)/2 + 1 ) is divisible by m. They are relatively prime, since these are successive odd integers. Suppose that m is a odd prime such that and (2^(m-1)/2 - 1 )/m is a square.

First, let us suppose that (2^(m-1)/2 - 1 ) is divisible by m. We know that, if x and y are relatively prime positive integers such that their product xy is a perfect square, then x and y separately must be perfect squares.

Since, (2^m -1 )/m = ((2^(m-1)/2 - 1 )/p)(2^(m-1)/2 + 1 ); we observe that 2^(m-1)/2 + 1 = a^2, for some integer a gt 2.

Thus; 2^(m-1)/2 = a^2 - 1 = (a-1)(a+1)
If a-1 = 2^r, a+1 = 2^s for 0 lt r lt s, then:
2^s - 2^r = 2, giving 2^(s-1) - 2^(r-1) = 1
Since, 0 lteq (r-1) lt (s-1), this is possible only if r-1=0, s-1 = 1.
Hence, a-1 =2, so that a=3;
2^(m-1)/2 = a^2-1 = 8; (m-1)/2 =3, giving m = 7

Next, let us suppose that 2^(m-1)/2 + 1 is divisible by m.

Since, (2^m -1 )/m = (2^(m-1)/2 - 1 )(2^(m-1)/2 + 1 ); we observe, exactly as before that 2^(m-1)/2 - 1 =b^2, for some integer b gt 0.

If, (m-1)/2 gteq 2, then this implies that b^2 = -1 (Mod 4), which is not feasible.

Hence, (m-1)/2 lteq 1, so that m = 3, since m must be odd.

Finally, if m =7, then :
(2^(m -1)/2 - 1 )/m = 9 ,

And if m =3, then : (2^(m -1)/2 - 1 )/m = 1.

THUS, THE DESIRED PRIMES ARE 3 and 7.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtstypos re a solutionDennis2006-09-24 18:08:56
Solutiona solutionDennis2006-09-24 15:40:02
Some Thoughtsfermats little theoremsiddhesh2006-09-23 19:50:45
re: prelimanary findingsDej Mar2006-09-23 16:54:28
Hints/Tipsprelimanary findingsCharlie2006-09-23 16:05:01
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