262 Jean-Marie De Koninck

481 899

• the

12th

and largest known number n 1 such that n ·

2n

+ 1 is prime (see the

number 141).

485 475

• the smallest number n such that (n +

j)4

+ (n + j +

1)4,

for j = 0, 1, 2, 3, 4, are

all primes: here these primes are 111096215892626372264401,

111097131255474983589617, 111098046623980093284097,

111098961998141724650737 and 111099877377959900992481.

491 531

• the largest prime number p

232

such that

7p−1

≡ 1 (mod

p2):

the only

other known prime number p satisfying this property is p = 5.

505 925

• the largest known solution of γ(n + 1) − γ(n) = 17 (see the number 1 681).

507 904

• the

19th

Granville number (see the number 126).

509 203

• the largest known Riesel number: an odd number k is called a Riesel number

if k · 2n − 1 is composite for each number n ≥ 1: H. Riesel [172] proved that

k = 509 203 satisfies this property182 (see also the number 78 557).

512 000

• the smallest number n such that 10! divides 1 + 2 + . . . + n (see the number

224).

514 229

• the ninth prime Fibonacci number (see the number 89).

522 752

• the seventh solution of σ(n) = 2n + 2 (see the number 464).

182Riesel also proved that there exist infinitely many Riesel numbers: he did this by showing that

the numbers 509203 + 11184810r, r = 0, 1, 2, . . ., are all Riesel numbers. As of May 2009, we know

that the smallest number k which could possibly be a Riesel number is k = 2 293.