Those Fascinating Numbers 39

120

• the smallest tri-perfect number (n is tri-perfect if σ(n) = 3n): only six tri-

perfect numbers are known, namely 120, 672, 523 776, 459 818 240, 1 476 304 896

and 51 001 180 160, and it seems that there are no others (see R.K. Guy [101],

B2);

• the only number n which can be joined with the numbers 1, 3 and 8 to form the

set A = {1, 3, 8, n} so that if x, y ∈ A, x = y, then xy + 1 is a perfect square;

this result was obtained by Euler; for a thorough analysis of this problem, see

L. Jones [113] or the more recent

paper47

of Dujella [73];

• one of the five numbers (the others being 1, 10, 1 540 and 7 140) which are both

triangular and tetrahedral (see the number 10);

• the smallest solution of σ2(n) = σ2(n + 10): the only solutions n 108 of this

equation are 120, 942, 5 395, 4 737 595, 6 811 195, 11 151 355, 74 699 995 and

98 600 035.

121

• the smallest number n 1 which is

both48

a star number and a perfect square:

a star number is a number of the form 6n(n + 1) + 1; the sequence of num-

bers satisfying this property begins as follows: 121, 11881, 1164241, 114083761,

11179044361, 1095432263641, . . . ;

• the only known perfect square of the form 1+p + p2 + p3 + p4, where p is prime:

here with p = 3.

122

• the only known number whose square is the sum of a fourth power and a fifth

power: here

1222

=

114

+

35

(see H. Darmon & A. Granville [42] as well as

the number 21 063 928); in fact it is conjectured that the only co-prime integer

solutions x, y, z (non zero) of the equation

xp+yq

=

zr

, with

1

p

+

1

q

+

1

r

1, where

exactly49 one of the numbers p, q, r is equal to 2, are those appearing in the

47A

set of m positive integers {a1, a2, . . . , am} is called a diophantine m-tuple if aiaj + 1 is a

perfect square for all 1 ≤ i j ≤ m. The first diophantine quadruplet, that is {1, 3, 8, 120},

was found by Fermat. In 1969, Baker & Davenport [11] proved that this quadruplet could not

be extended to a diophantine quintuplet. Let us mention that in 1979, Arkin, Hoggatt & Strauss

[7] proved that each diophantine triplet could be extended to a diophantine quadruplet: indeed,

if {a, b, c} is such a triplet and if ab + 1 =

r2,

ac + 1 =

s2

and bc + 1 =

t2,

where r, s, t are

positive integers, then one easily verifies that d = a + b + c + 2abc + 2rst is such that {a, b, c, d}

is a diophantine quadruplet. In 2004, Dujella [73] proved that no diophantine 6-tuple exists and

that there can only exist a finite number of diophantine 5-tuples, and in fact that any element of a

diophantine 5-tuple must be smaller than

101026

. Let us add that it is easy to prove that there exist

infinitely diophantine quadruplets; indeed, one only needs to prove that there exist infinitely many

diophantine triplets and to use the result of Arkin, Hoggatt & Strauss mentioned above; one then

only needs to verify that the triplets {1,

r2

− 1,

r2

+ 2r}, where r = 2, 3, 4, . . ., are all diophantine.

48One can establish the recurrence formula Ek = 98Ek−1 − Ek−2 + 24, where Ek stands for the

kth

number which is both a star number and a perfect square.

49According to the Beal Conjecture, there are no solutions with min(p, q, r) ≥ 3.