344 Jean-Marie De Koninck

3 215 031 751

• the smallest strong pseudoprime in bases 2, 3, 5 and 7.

3 262 811 042

• the smallest number (and possibly the only one202) which can be written as

the sum of the fourth powers of two prime numbers in two distinct ways:

3 262 811 042 = 74 + 2394 = 1574 + 2274 (see the numbers 635 318 657 and

6 058 655 748).

3 387 574 069

• the second number n such that

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4))

5

4

:

here

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4))

≈ min(1.30231, 1.25716, 1.25552, 1.29489, 1.30511) = 1.25552;

(see the number 14 018 750).

3 413 246 929

• the third powerful number which can be written as the sum of two co-prime 4-

powerful numbers: 3 413 246 929 = 4 170 272+3 409 076 657, that is 372 ·15792 =

25 · 194 + 74 · 175 (see the number 12 769).

3 504 597 120

• the

13th

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

202Here

is the heuristic argument in favor of the fact that 3 262 811 042 is possibly the only number

satisfying this property. Indeed, first of all it is clear that the number of numbers n ≤ x of the form

p4

+

q4

is ≈

x1/4

log x

2

=

x1/2

log2

x

. This is why one can claim that the probability that a number

n chosen at random is of the form n =

p4

+

q4

is

≈

n1/2/ log2

n

n

=

1

n1/2

log2

n

.

It follows that Prob[n = p1

4

+ q1

4

= p2

4

+ q2

4]

≈

1

n1/2

log2

n

2

=

1

n

log4

n

, so that the expected

number of integers n such that r ≤ n ≤ x having two such representations is

≈

r≤n≤x

1

n

log4

n

≈

x

r

1

t

log4

t

dt = O

1

log3

x

+ O

1

log3

r

= O

1

log3

r

,

which tends to 0 as r becomes large.