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.
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