Those Fascinating Numbers 23

• the smallest bizarre number: we say that a number n is bizarre if it is abundant

(that is if σ(n) 2n) and non pseudo-perfect (see the number 12): the sequence

of bizarre numbers begins as follows: 70, 836, 4030, 5830, 7192, 7912, 9272,

10430, . . .

71

• the only prime number p 232 such that 11p−1 ≡ 1 (mod p2) (see Ribenboim

[169], p. 347);

• the second prime number q which divides the sum of all prime numbers q

(that is q|

∑

pq

p): this sequence of prime numbers begins as follows: 5, 71,

369 119, 415 074 643, . . . ;

• the largest positive solution y of x(x + 1) . . . (x + 5) = y2 − 1: this solution is

(x, y) = (2, 71) (see R.K. Guy [101], D25); in fact, P.T. Bateman [17] proved

that the only solutions of this diophantine equation, other than the trivial

solutions x = −5,−4,−3,−2,−1, 0 and y = ±1, are (x, y) = (−7, ±71) and

(2, ±71); see also the number 142;

• the largest known solution m of n! + 1 =

m2,

here with n = 7;

• the largest known number k such that the decimal expansion of

2k

does not

contain the digit 5 nor the digit 7; indeed, using a computer, one can verify

that

271

= 2 361 183 241 434 822 606 848

does not contain the digit 5, nor the digit 7, while each number 2k, for k =

72, 73, . . . , 3000, contains these two digits (see David Gale [86]) 31.

which indeed represents an elliptic curve on which we are searching for points with integer coor-

dinates. Some of these points are obvious, namely the points (0, 0), (−1, 0), (−

1

2

, 0), (1, 1) and

(1, −1). With the technique used to add two points on an elliptic curve (see for instance the book

of Larry Washington [199]), we obtain

((0, 0) + (1, 1)) + (1, 1) =

1

2

, −

1

2

+ (1, 1) = (24, −70).

We may therefore conclude that the point (x, y) = (24, −70) is indeed a point on the elliptic curve

y2

=

x3

3

+

x2

2

+

x

6

. As a final note, let us mention that no one has yet proved, using the theory

of elliptic curves, that it is indeed the only point with integer coordinates, except of course for the

trivial ones mentioned above.

31The

largest known numbers k such that the corresponding number

2k

does not contain the

digit , for = 0, 1, 2, . . . , 9, are respectively 86, 91, 168, 153, 107, 71, 93, 71, 78 and 108; using a

computer, one can easily verify this statement for each number

2k

with k 3000. Moreover, from

a probabilistic point of view, it follows from this observation that the probability that there exists

a number 2k with k 3000 not containing a given digit is of the order of 10−136. Indeed, it is

clear that the probability that a number chosen at random amongst all those numbers with r digits

does not contain a given digit ∈ [0, 9] is equal to

8

9

×

9

10

r−1

. Now, the number of powers of

2 with r digits is approximately

log 10

log 2

. This is why the probability that there exists a power of 2

with r digits, r 3000, not containing the digit , for a certain ∈ [0, 9], is approximately

r3000

log 10

log 2

×

8

9

×

9

10

r−1

≈

10−136.