Pn=n(3n-1)/2. The first ten pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ... It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 - 22 = 48, is not pentagonal. Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference are pentagonal and D = |Pk - Pj| is minimised; what is the value of D? """ from Helper import isPentagonal import sys def pentagonal(n): return n*(3*n-1)/2 k = 2 minD = 10**10 while True: for j in range(k-1, 0, -1): pk = pentagonal(k) pj = pentagonal(j) if pk - pj < minD: if isPentagonal(pk-pj) and isPentagonal(pk+pj): minD = pk-pj sys.exit(0) k += 1
""" Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, ... Pentagonal Pn=n(3n-1)/2 1, 5, 12, 22, 35, ... Hexagonal Hn=n(2n-1) 1, 6, 15, 28, 45, ... It can be verified that T285 = P165 = H143 = 40755. Find the next triangle number that is also pentagonal and hexagonal """ from Helper import isPentagonal, isHexagonal n = 286 while True: tn = n*(n+1)/2 if isPentagonal(tn) and isHexagonal(tn): print tn break n += 1
