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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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Solution Program Solution | Comment 3 of 10 |

I took the arithmetic sequence as q-d, q, q+d.  Then N^3-1=(q-d)*q*(q+d) and d=sqrt(q^2-(N^3-1)/q)

q must be at least as large as N, otherwise the product would be too small because q+d could not be large enough to compensate for q and q-d less than N. Similarily, q must be less than N^1.5 because if it was larger then q*(q+d) would already exceed N.

Then I wrote a short UBASIC program which looped N=2 to 201, and for each N looped q from N to n^1.5:

10 for n=2 to 201
19 qmax=int(n^1.5)
20 q=n to qmax
30 if (n*n*n-1)@q<>0 then 70
40 x=q*q-(n*n*n-1)/q
41 d=isqrt(x)
50 if d*d=x then print n,q-d;q;q+d
70 next q
80 next n
Running this short program finds:
9       2  14  26
25      2  63  124
49      2  172  342
81      2  365  728
100     27  143  259
121     45  152  259
121     28  185  342
121     2  666  1330
169     2  1099  2196
196     117  211  305

There are 8 distinct values for N: 9,25,49,81,100,121,169,196 for a probability of 1/25.  Continuing the program finds that not all N are perfect squares.  N=273 is an answer with sequence 38, 527, 1016.


  Posted by Brian Smith on 2010-06-01 01:38:21
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