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Infinite Triplet and Perfect Square (Posted on 2015-08-16) Difficulty: 3 of 5
Does there exist an infinite number of positive integer triplets (x,y,z) with x < y < z such that:
x, y and z describe an arithmetic sequence, and:
Each of xy+1, yz+1 and zx+1 is a perfect square?

Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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a few more | Comment 2 of 3 |
Starting with Jer's three solutions, I note that the z = 2y-x.
Also that x = previous y/2 and y = previous z*2

That makes the next few solutions of this form
x       y       z
1 8 15
4 30 56
15 112 209
56 418 780
209 1560 2911
780 5822 10864
2911 21728 40545
10864 81090 151316
40545 302632 564719
151316 1129438 2107560
564719 4215120 7865521
2107560 15731042 29354524
I have generated these using Excel and checked using Excel that these are all valid solutions.  And the first column is OEIS A001353

Thanks for the start, Jer.

Edited on August 16, 2015, 4:11 pm
  Posted by Steve Herman on 2015-08-16 16:07:44

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