312 Jean-Marie De Koninck

43 112 609

• the exponent of the 46th known Mersenne prime 243 112 609 −1 (a 12 978 189 digit

number) discovered by Edson Smith in September 2008, using the programme

developed by G. Woltman (see the number 1 398 269)193.

43 570 803

• the smallest number n such that φ(n), φ(n + 1), φ(n + 2), φ(n + 3), φ(n + 4)

and φ(n + 5) have the same prime factors, namely here the primes 2, 3, 5, 7,

13 and 19 (see the number 266 401; see also the number 3 777 for the analogue

problem for the σ(n) function).

45 086 079

• the number of nine digit prime numbers (see the number 21).

45 326 160

• the fifth number n such that φ(n) + σ(n) = 4n (see the number 23 760).

45 532 800 (=

27

·

33

·

52

· 17 · 31)

• the fifth 4-perfect number (see the number 30 240): Carmichael proved that it

is the largest 4-perfect number with exactly five distinct prime factors.

45 581 759

• the

11th

solution of σ2(n) = σ2(n + 2) (see the number 1 079).

45 592 577

• the smallest prime factor of F10 =

2210

+ 1, whose complete factorization is

given by

F10 = 45592577 · 6487031809

·4659775785220018543264560743076778192897 · P252.

45 593 065 (=5 · 7 · 17 · 19 · 37 · 109)

• the smallest pseudoprime in base 2 having exactly six prime factors (see the

number 11 305).

193Oﬃcially, the discoverer received a 100 000$ reward. A 150 000$ prize will be awarded for the

discovery of a 100 million digit prime number; and we might have to wait some time before someone

wins the 250 000$ prize for the discovery of a billion digit prime number.