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Near Fermat Formulation (Posted on 2015-05-11) Difficulty: 3 of 5
Prove that the equation:
AN + BN = CN-1
has an infinity of positive integer solutions for every integer value of N ≥ 2

No Solution Yet Submitted by K Sengupta    
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re(2): solution | Comment 3 of 4 |
(In reply to re: solution by JayDeeKay)

good catch, missed the part about infinite solutions.


A=B=2^p
then we have
A^n+B^n=
2*[2^p]^n
2*2^(pn)
2^(pn+1)
so for this to be a perfect (n-1) power we need
pn+1 = 0 mod (n-1)
but n = 1 mod (n-1) thus this reduces to
p+1 = 0 mod (n-1)
p = -1 mod (n-1)
p = n-2 mod (n-1)
p=(n-1)k+n-2
p=kn-k+n-2
p=(k+1)n-k-2

thus for any given N we have a family of solutions:
p=(k+1)n-k-2
A=B=2^p
C=2^[(pn+1)/(n-1)]
for all k>=0

  Posted by Daniel on 2015-05-12 04:45:21
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