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Cubic and Consecutive Concern II (Posted on 2010-05-31) Difficulty: 3 of 5
Determine the probability that for a positive base ten integer N drawn at random between 2 and 201 inclusively, the number N3 - 1 is expressible in the form p*q*r, where p, q and r are three distinct positive integers such that p, q and r (in this order) corresponds to three consecutive terms of an arithmetic progression.

No Solution Yet Submitted by K Sengupta    
Rating: 3.5000 (2 votes)

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Some Thoughts Infinite family of solutions | Comment 8 of 10 |

One striking thing about the solutions is that so many of them have p=2.  Directly substituting p=2 into the equations makes r=2q-2 and then N^3=(2q-1)^2.  These must both equal a number to the sixth power, call it k.  2q-1 is odd, therefore k and N are odd.  Then for every odd integer k, there is a solution N=k^2, p=2, q=(k^3+1)/2, r=k^3-1

Note: this does not cover all solutions, just ones with p=2.

Edited on June 1, 2010, 8:23 pm
  Posted by Brian Smith on 2010-06-01 20:21:58

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