Undulating number

• There are infinitely many undulating numbers. • For any n ≥ 3, there are 9 × 9 = 81 non-trivial n-digit undulating numbers, since the first digit can have 9 values (it cannot be 0), and the second digit can have 9 values when it must be different from the first. • Every undulating number with even number of digits and at least four digits is composite, since: ABABAB...AB = 10101...01 × AB. For example, 171717 = 10101 × 17. • Undulating numbers with odd number of digits are palindromic. They can be prime, for example 151. • The undulating number ABAB...AB with n repetitions of AB can be expressed as AB × (102n − 1)/99. For example, 171717 = 17 × (106 − 1)/99. • The undulating number ABAB...ABA with n repetitions of AB followed by one A can be expressed as (AB × 102n+1 − BA)/99. For example, 989898989 = (98 × 109 − 89)/99 • Undulating numbers can be generalized to other bases. If a number in base b with even number of digits is undulating, in base b^2 it is a repdigit. == Undulating primes ==