# powerful number

A positive integer *n* is **powerful** if for every prime *p*
dividing *n*, *p*^{2} also divides *n*. The reader
might want to pause and show that the powerful numbers are exactly those that can
be written *a*^{2}*b*^{3}, 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
PandQwithPeven andQodd such thatP^{2}−Q= 1.- There exists a square free positive integer
m≡ 7 (mod 8) with T_{1}+U_{1}·√mbeing the fundamental unit of Q*(√m) and, for some odd integerk, T_{k}is even powerful and U_{k}≡ 0 (modm) is an odd number where(T

_{1}+ U_{1}·√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. MollinandP. Walsh, "On powerful numbers,"Intern. J. Math. Math. Scu.,9(1986) 801--806.MR 88f:11005

Printed from the PrimePages <t5k.org> © Reginald McLean.