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Cubic and Consecutive Concern (Posted on 2010-03-11) Difficulty: 2 of 5
Determine the probability that for a base ten positive integer N drawn at random between 2 and 101 inclusively, the number N3 + 1 is expressible in the form p*q*r, where p, q, r are three distinct positive integers such that p, q and r (in this order) corresponds to three consecutive terms of an arithmetic sequence.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2

The probability is 5%, as there are only 5 values of n that work: 3, 12, 27, 48 and 75.

 start diff   seq.      product  n-formula
     1   3:  1  4  7         28   3^3+1
     7   6:  7 13 19       1729  12^3+1
    19   9: 19 28 37      19684  27^3+1
    37  12: 37 49 61     110593  48^3+1
    61  15: 61 76 91     421876  75^3+1
   

It was more efficient to try all possible arithmetic sequences rather than trying all possible n's and trying to factor the enhanced cubes. The starting number in the sequence couldn't be higher than 101, but 102 was used for good measure. The difference between terms of the sequence couldn't be higher than 1015 as higher would exceed 101^3+1 even if starting at 1. When starting higher than 1, the inner loop is exited as soon as products start to exceed the limit.  
   
DEFDBL A-Z
DIM yes(101)
limit = 101 ^ 3 + 1
FOR st = 1 TO 102
  FOR diff = 1 TO 1016
     prod = 1
     FOR i = st TO st + 2 * diff STEP diff
       prod = prod * i
     NEXT
     IF prod > limit THEN EXIT FOR
     cube = prod - 1
     cr = INT(cube ^ (1 / 3) + .5)
     IF cr * cr * cr = cube AND cr > 1 AND cr < 102 THEN
       PRINT USING "### ###: ## ## ##    ####### ###&"; st; diff; st; st + diff; st + 2 * diff; cube + 1; cr; "^3+1"
       yes(cr) = 1
     END IF
  NEXT diff
NEXT st
tot = 0
FOR i = 2 TO 101
  tot = tot + yes(i)
NEXT
PRINT tot


  Posted by Charlie on 2010-03-11 13:45:10
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