318 Jean-Marie De Koninck
87 539 319
the smallest number which can be written as the sum of two cubes in three dis-
tinct ways: 87 539 319 = 1673 +4363 = 2283 +4233 = 2553 +4143 (Leech, 1957):
the sequence of numbers satisfying this property begins as follows: 87 539 319,
119 824 488, 143 604 279, 175 959 000, 327 763 000, . . . ; see the number 1 729 as
well as R.K. Guy [101], D1.
87 699 842
the smallest Niven number n such that n + 100 is also a Niven number, but
with no others in between (see the number 28 680).
88 593 477
the largest number equal to the sum of the eighth powers of its digits (see the
number 24 678 050).
88 612 802
the smallest number which can be written as the sum of two, three and four
distinct fourth powers: indeed, 88 612 802 = 174 + 974 = 214 + 804 + 834 =
254 + 264 + 604 + 934.
89 002 914
the tenth number n such that β(n)|β(n +1) and β(n +1)|β(n +2), here 138|966
and 966|966 (see the number 225 504).
89 351 671
the only prime number p
232
such that
66p−1
1 (mod
p2)
(P.L. Montgo-
mery [143]).
89 397 016
the 15th dihedral perfect number (see the number 130).
89 460 294
the smallest number n (and the only one 1011) such that (∗) β(n) = β(n +
1) = β(n + 2); here n = 2 · 3 · 7 · 11 · 23 · 8419, n + 1 = 5 · 4201 · 4259 and
n + 2 =
23
· 31 · 43 · 8389, so that β(n) = β(n + 1) = β(n + 2) = 8465; although
n = 89460294 is the only known solution of (∗), one may be lead to believe that
(∗) has infinitely many
solutions195
(see the number 417 162 for the analogue
problem when one replaces β(n) by B(n), as well as the number 15 860 for the
similar problem when β(n) is replaced by β∗(n)).
195In fact, if C(x) stands for the number of solutions n x of (∗), Carl Pomerance (see J.M. De
Koninck [46]) provided a heuristic argument suggesting that not only C(x) as x ∞, but
that, given any real number ε 0, C(x) x1−ε for all x sufficiently large.
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