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 Expression and Perfect Square 2 (Posted on 2015-12-25)
Each of A,B and C is a positive integer such that each of the expressions A*B+A+B, B*C+B+C, C*A+C+A, A*B+C and A*C+B is a perfect square.

(I) Find the smallest value of A+B+C.

(II) Does there exist an infinity of triplets (A,B,C) satisfying the given conditions?

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (2 votes)

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 re: Two small solutions and a conjecture | Comment 2 of 3 |
(In reply to Two small solutions and a conjecture by Brian Smith)

After some more exploration I have come up with this conjecture:

Any consecutive three terms of OEIS A027941 (0, 1, 4, 12, 33, 88, 232, ...) form a triplet ABC which satisfies the given problem.

Let F(n) be the nth term of the Fibonacci sequence.  Then for parameter k:
A = F(2k-1)-1
B = F(2k+1)-1
C = F(2k+3)-1
A*B+A+B = F(2k)^2
A*C+A+C = F(2k+1)^2
B*C+B+C = F(2k+2)^2
A*B+C = [F(2k)+1]^2
A*C+B = [F(2k+1)-1]^2
B*C+A = [F(2k+2)-1]^2

 Posted by Brian Smith on 2015-12-26 13:50:47

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