Those Fascinating Numbers 209

30 618

• possibly the largest number n such that n(n + 1)(n + 2) . . . (n + 6) and (n +

1)(n + 2) . . . (n + 7) have the same prime factors: here

30618 · 30619 · . . . · 30624 =

29

·

39

· 5 · 7 · 11 · 29 · 59 · 61

·67 · 113 · 173 · 251 · 271 · 457 · 1531,

30619 · 30620 · . . . · 30625 =

28

·

32

·

55

·

72

· 11 · 29 · 59 · 61

·67 · 113 · 173 · 251 · 271 · 457 · 1531;

if nk, for k ≥ 2, stands for the largest number n such that n(n + 1)(n +

2) . . . (n + k − 1) and (n + 1)(n + 2) . . . (n + k) have the same prime factors,

then the conjectured values of the first nk’s (assuming the abc

Conjecture160)

are n2 = 2, n3 = 24, n4 = 32, n5 = 400, n6 = 480, n7 = 30 618, n8 = 34 992,

n9 = 39 366, n10 = 43 740 and n11 = 107 800.

30 693

• the ninth number which is not a palindrome, but whose square is a palindrome

(see the number 26).

30 784

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

powers: 30 7842 = 964 + 1204 + 1604 (see the number 481).

31 469

• the smallest prime number which is preceded by 71 composite numbers; indeed,

there are no primes between 31 397 and 31 469.

31 907

• the smallest prime number p such that p + 50 is prime and such that each

number between p and p + 50 is composite (see the number 139).

160Indeed, it is easy to show that if the abc Conjecture is true, then each number nk is well defined.

Indeed, let k ≥ 2 be fixed. First observe that P (n) ≤ k, since otherwise p|n for some prime p k,

in which case p cannot divide any of the numbers n + i for i = 1, 2, . . . , k, thereby contradicting the

fact that n(n + 1) . . . (n + k − 1) and (n + 1) . . . (n + k) have the same prime divisors. By the same

argument, we also have P (n + k) ≤ k. Thus, applying the abc Conjecture (with a = n, b = k and

c = n + k), we have

n + k γ(n · k · (n +

k))1+ε

p≤k

p

1+ε

,

an inequality which cannot hold if n is suﬃciently large.