perplexus.info :: Just Math : Near Fermat Formulation

Near Fermat Formulation (Posted on 2015-05-11)

Prove that the equation:
A^N + B^N = C^N-1
has an infinity of positive integer solutions for every integer value of N ≥ 2

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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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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