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I found that the number : $$123456789101112131415161718192021222324252627282931$$ which is prime number , I want to know if there is any simple method to show students its primality ?

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For high school level, showing non-primality is fair game, provided simple divisibility criteria can resolve it.

Showing primality for a number such as the one you posted is definitely not fair game.

One guideline would be: If the teacher can't show it easily, then it's not appropriate for students.

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The number is a prime. This cannot be found out in reasonable time without a computer.

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    $\begingroup$ To back this claim up: the largest known primes before the advent of the electronic computer give an indication of what's feasible to do by hand. All of them are essentially Mersenne primes and largest of them is a mere $\approx 10^{38}$. OP's prime is not a Mersenne prime and is $\approx 10^{50}$ so it's very unlikely to have an easy proof. $\endgroup$ – Jam Jul 17 at 17:09
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This isn't practical. Without knowing every prime below about $10^{25}$ you can't do simple divisibility testing (even if you could do it all before the universe ends). Even sieving with a $$6k\pm1$$ sieve adapted from the sieve of sundaram you would either need to type out more than $10^{24}$ numbers and sieve them as you go, just to get base primes. Primorial base change only helps to a point. Beyond which a primality check of some form must be known. Preferable a fast one.

For a comparison , I wrote out all numbers up to 800 ( might go a bit further) but even if I do a sieve based on form, I'll only get all primes under about 4900 back. it's taken me about a week on and off. Yours would take a lifetime. Have fun trying to teach primality checks. even mod 30 we have 6 equations for it's mod 30 class to check.

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