Those Fascinating Numbers 89

470

• the fifth number α such that σ(n) =

2α

for all numbers n; the sequence of

numbers α satisfying this property begins as follows: 1, 4, 6, 11, 470, 475, 477,

480, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496,

497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512,

513, 514, 515, 516, 517, 518, 519, 520, 522, 525, 527, 532, . . . 92

479

• the smallest prime factor of the Mersenne number

2239

− 1, whose complete

factorization is given by

2239

− 1 = 479 · 1913 · 5737 · 176383 · 134000609

·7110008717824458123105014279253754096863768062879.

481

• the smallest number whose square can be written as the sum of three fourth

powers:

4812

=

124+154+204;

the ten smallest numbers satisfying this property

are 481, 1 924, 4 329, 7 696, 12 025, 17 316, 23 569, 24 961, 28 721 and 30 784;

• possibly the largest number n such that n(n + 1) . . . (n + 5) has exactly the

same prime factors as m(m + 1) . . . (m + 5) for a certain number m n: here

m = 480 and the prime factors common to these two quantities are 2, 3, 5, 7,

11, 13, 23, 37, 97 and 241, since

480 · 481 · . . . · 485 =

28

·

32

·

52

· 7 ·

112

· 13 · 23 · 37 · 97 · 241,

481 · 482 · . . . · 486 =

24

·

36

· 5 · 7 ·

112

· 13 · 23 · 37 · 97 · 241;

see the number 340.

485

• the fifth number n such that n2 − 1 is powerful: here 4852 − 1 = 23 · 35 · 112; the

sequence of numbers satisfying this property is infinite93 and begins as follows:

3, 17, 26, 99, 485, 577, 1 351, 3 363, 19 601, 24 335, 70 226, 114 243, 470 449,

665 857, 930 249, 2 862 251, 3 650 401, 3 880 899, . . .

487

• the second prime number p such that

10p−1

≡ 1 (mod

p2):

the only prime

numbers p

232

satisfying this congruence are 3, 487 and 56 598 313 (see

Ribenboim [169], p. 347).

92It

is easy to prove that σ(n) is a power of 2 if and only if n is the product of Mersenne primes.

93This follows from the fact that to each solution (x, y) of the Fermat-Pell equation x2 − 2y2 = 1,

one can associate a number n such that

n2

− 1 is powerful. Indeed, choosing n = x, we then have

n2 − 1 = x2 − 1 = 2y2, and since y is necessarily even, it follows that 2y2 is powerful.