powerful number

A positive integer n is powerful if for every prime p dividing n, p² also divides n. The reader might want to pause and show that the powerful numbers are exactly those that can be written a²b³, where a and b are positive integers. Here are the powerful numbers up to 1000:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, and 1000.

There are pairs of consecutive powerful numbers such as:

(8,9), (288,289), (675,676), (9800,9801), (12167,12168), (235224,235225), (332928,332929) and (465124,465125).

Erdös conjectured in 1975 that there do not exist three consecutive powerful integers. Golomb also considered this question in 1970, as did Mollin and Walsh (independently) in 1986. The latter proved that the following are equivalent:

• There EXIST three consecutive powerful numbers.
• There exist powerful numbers P and Q with P even and Q odd such that P² − Q = 1.
• There exists a square free positive integer m ≡ 7 (mod 8) with T₁+U₁·√m being the fundamental unit of Q*(√m) and, for some odd integer k, T_k is even powerful and U_k ≡ 0 (mod m) is an odd number where

  (T₁ + U₁·√m)^k = T_k + U_k·√m.

References:

Erdos1975

P. Erdös, "Problems and results on consecutive integers," Eureka, 38 (1975/6) 3--8.  MR 56:11931

Erdos1976

P. Erdös, "Problems and results on consecutive integers," Publ. Math. Debrecen, 23 (1976) 271--282.  MR 56:11931

MW1986

R. Mollin and P. Walsh, "On powerful numbers," Intern. J. Math. Math. Scu., 9 (1986) 801--806.  MR 88f:11005