Determine all possible values of a positive integer N ≥ 3, such that ^NC₂ – 1 is a prime power.

Note: ^NC₂ represents N choose 2.

list
   10       N=3:PrevN=2
   20       while N>=3
   30         C=N*PrevN//2-1
   40         D=prmdiv(C)
   50         C=C//D:Ct=1
   60         while C>1
   70          DNew=prmdiv(C)
   80          if DNew<>D then goto *NotGood
   90          C=C//D:inc Ct
  100         wend
  110         print N,N*PrevN//2-1,D;Ct
  120    *NotGood
  130         PrevN=N
  140         inc N
  150       wend
OK
run
 3       2       2  1
 4       5       5  1
 5       9       3  2
 8       27      3  3
Division by zero in 50
?n,c
 262260          34390022669
OK

The program stopped at N = 262,260, when the combination less 1 became too big (34,390,022,669) for UBASIC to find its lowest prime divisor.

The table is interpreted:

 N   C(N,2)-1   equals
 3       2       2^1
 4       5       5^1
 5       9       3^2
 8       27      3^3
 

and no other value of N lower than 262,260 will work.

Posted by Charlie on 2010-04-18 13:45:26