Glitch Prime

Number Theory Level 3

A recent Numberphile (a math YouTube Channel) video is about Glitch Primes. (

A Glitch Prime is a number in which can be written in the form of \(10^{x} - 10^{y} - 1\) where \(x=2y\). This number should be a palindrome number (same forwards as backwards) and only consists of two different numbers. Finally, this number must be a prime number.

Let me give you an example: \(10^{2} - 10^{1} - 1 = 89\) (in base ten). I mentioned that this is in base 10 because there are other glitch primes in different bases (like 101 in base 2).

Ok, now I am going to tell you to ignore the fact that these should be prime numbers. Now, calculate the first non-prime glitch primes for bases 2-16, then convert these numbers into decimal. Upon viewing the last digit in each of these numbers, you will notice a pattern that repeats exactly 3 times in bases 2 to 16.

Take 1 repetition of this pattern and mush all of the digits together (for example, if you had 4, 7, 2, 5, then write it as 4725). Type this number as your answer.

Note: I may have made a mistake when typing up this question. If you notice a mistake or something that is unclear, please let me know in the "Report a Problem" section and I will try to fix it.


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