A sequence {B_p} of positive integers is such that:

B₁ = 20, B₂ = 30, and:

B_p+2 = 3*B_p+1 – B_p, whenever p ≥ 1.

Determine all possible positive integer values of p such that:
1 + 5*B_p+1*B_p is a perfect square.

**** For an extra challenge, derive a non computer program assisted solution.

   10     B1=20:B2=30
   20     Ct=1
   30
   40     while B3<10^2000
   50       Sq=1+5*B2*B1
   60       Sr=int(sqrt(Sq)+0.5)
   70       if Sr*Sr=Sq then print Ct,B1;B2,Sq;Sr
   80       B3=3*B2-B1
   90       B1=B2:B2=B3
  100       Ct=Ct+1
  110     wend
OK
run
 3       70  180         63001  251
Overflow in 60
?ct
 3075
 
After checking over 3000 iterations, only 70 and 180, the next two numbers after 20 and 30 satisfy the square condition, with the formula producing 1 + 5*70*180 = 63001 = 251^2.

Posted by Charlie on 2013-12-09 16:43:04