Is there any proof that there are only

Andreas Gallegos

Andreas Gallegos

Answered question

2022-04-14

Is there any proof that there are only 3 so-called "neon numbers"?

Answer & Explanation

folytonr4qx

folytonr4qx

Beginner2022-04-15Added 14 answers

Step 1
Suppose a number m has n digits. Then m2 has atmost 2n digits.
The sum of digits of m2 is at most 9×2n=18n
If the sum of digits of m2 is equal to m then
10n1m18n
This only true if n<3, so only single digit and double digit numbers can possibly be neon numbers.
Step 2
You can prove using modular arithmetic that only numbers of the form 9k and 9k+1 can be neon numbers. This reduces the number of positive integers that we have to check.

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