Those Fascinating Numbers 303
20 291 270
the sixth number n such that Eσ(n) := σ(n + 1) σ(n) satisfies Eσ(n + 1) =
Eσ(n): here the common value of is 365 040, since σ(20291270) = 37374480,
σ(20291271) = 37739520 and σ(20291272) = 38104560 (see the number 693).
20 427 264 (= 29 · 32 · 11 · 13 · 31)
the fourth solution of
σ(n)
n
=
7
2
(see the number 4 320).
20 511 392
the smallest number which can be written as the sum of two fifth powers and
as the sum of four fifth powers: 20 511 392 = 35 + 295 = 45 + 105 + 205 + 285.
20 831 323
the smallest prime number which is followed by at least 200 consecutive compos-
ite numbers (in fact here by exactly 209 composite numbers); see the number
370 261.
20 840 574
the third solution of σ(n) = σ(n + 15) (see the number 26).
20 916 224
the smallest number n such that 12! divides 1 + 2 + . . . + n (see the number
224).
20 996 011
the exponent of the
40th
Mersenne prime
220 996 011
1 (a 6 320 430 digit num-
ber) discovered by Michael Shafer on November 17, 2003 using the programme
developed by G. Woltman (see the number 1 398 269).
21 063 928
the third number whose square can be written as the sum of a cube and a
seventh power (see B. Poonen, E. Schaefer & M. Stoll [165]): here 21 063 9282 =
762713 + 177; it is indeed possible to prove that the diophantine equation x2 =
y3 + z7 has only three solutions191 in positive integers x, y, z with (x, y, z) = 1;
here is the table of these solutions:
191In
1995, Henri Darmon and Andrew Granville [42] proved that if p, q, r are three positive integers
such that
1
p
+
1
q
+
1
r
1, then equation
Axp
+
Byq
=
Czr,
where A, B, C are non zero integers,
has only a finite number of solutions in positive integers x, y, z with (x, y, z) = 1.
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