MathDB

A positive integer

a

is called a double number if it has an even number of digits (in base 10) and its base 10 representation has the form

a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k

with

0 \le a_i \le 9

for

1 \le i \le k

, and

a_1 \ne 0

. For example,

283283

is a double number. Determine whether or not there are infinitely many double numbers

a

such that

a + 1

is a square and

a + 1

is not a power of

10

Problem Statement