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I am a person who gets bored all the time, so I found that counting something in my head helps occupy my time. I started counting in binary, but after a while this started getting old as I was doing it in the back of my mind, so I got bored again. So, my question is, can you count up prime numbers in your head using a "mind-friendly" formula, or is there something different you can count that keeps your mind busy?

Etscharntmin
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    No, there is not an easy pattern to find the next prime in sequence without a lot of work. At best, you can try to memorize tables of prime numbers as far as you can from what was published in the table, but we only know so many prime numbers so far (even though we know there must be infinitely many). That being said, reciting the prime numbers into the five-digits would already take a considerably long time and considerably strong memory and should more than suffice to fit your purpose. Alternatively, you could also try memorizing the first however many digits of $\pi$. – JMoravitz Sep 18 '23 at 14:48
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    If there were such a way , there would not be a largest known prime since we could immediately give a larger one. We cannot prove that there is no such way , but its existence is extremely unlikely , the existence of God is more likely although already incredibly unlikely. – Peter Sep 18 '23 at 15:06
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    For very small numbers , you can quickly see which of them are prime, but even for $4$ or $5$ digit-numbers , you virtually have to memorize them. You can try this , since other challenges are only (for whatever reason) made with the digits of $\pi$ , other equal interesting and important numbers like $e$ or $\sqrt{2}$ are apparently not memorized. – Peter Sep 18 '23 at 15:08
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    You could use the tests for divisibility here to check for divisors up to 30. If $n < 31 * 37$ and $d$ is not a divisor of $n$ for any $d \leq \min(30, \sqrt{n})$, then $n$ is prime. Good luck! – Jair Taylor Sep 18 '23 at 15:57
  • @Peter i have a somewhat of a pattern found here: https://math.stackexchange.com/questions/4772511/possibly-repeating-pattern-in-prime-numbers – Etscharntmin Sep 20 '23 at 14:42

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Here are a couple of ideas to mentally amuse yourself:

Prime related: Learn primes up to say $1000$ (or higher if you like). Then mentally recite the list backwards: $997, 991, ... $

Not prime related: Count backward from $3000$ by $37$'s ($3000, 2963,...$). Obviously you could pick a different starting value and a different increment when you get bored.

Honestly, it seems like there should be unlimited possibilities to keep oneself mentally occupied with numbers.

paw88789
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    Indeed , getting bored can also be avoided by learning the first $200$ digits of $e$ or any other irrational constant. – Peter Sep 18 '23 at 16:16
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    A better way is to count to $1000$, factoring non-prime numbers as you go. $1,2,3,2^2,5,2\cdot 3, 7\dots$ – John Douma Sep 18 '23 at 16:18
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You can try finding primes using Fermat's factorization method. You can find the squares by adding the odd numbers. You can combine the two and try finding prime numbers between two consecutive squares.

pietfermat
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For small numbers there are plenty of tricks. After the prime numbers 2 and 5 the rest end in 1,3,7,9. Checking divisibility by 3 is a snap. So, then we just need to exclude the products of $\{7,11,13,19\}$. That is just 6 numbers to track. And you have counted all of the primes up to 400.

Proceeding farther is more work, but you are looking for a mild mental challenge to keep yourself busy.

As others have pointed out, factoring large numbers is a “hard” problem.

user317176
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  • I tested this theory and found a pattern in larger numbers (only testing if they were divisible by 3). I discovered the following pattern: 1001: Prime 1003: Prime 1007: Prime 1009: Prime

    1011: Not a Prime 1013: Prime 1017: Not a Prime 1019: Prime

    1021: Prime 1023: Not a Prime 1027: Prime 1029: Not a Prime

    and then it just repeats:

    1031: Prime

     – Etscharntmin Sep 20 '23 at 13:26
  • Sorry cant edit my older comment, heres the finished one: I tested this theory and found a pattern in larger numbers (only testing if they were divisible by 3). I discovered the following pattern: 1001: Prime 1003: Prime 1007: Prime 1009: Prime

    1011: Not a Prime 1013: Prime 1017: Not a Prime 1019: Prime

    1021: Prime 1023: Not a Prime 1027: Prime 1029: Not a Prime

    and then it just repeats:

    1031: Prime 1033: Prime 1037: Prime 1039: Prime

    This is only abount ~60% correct due to only testing for multipiels of 3. Could the testing for 7, 11, 13, 19 also be repeatable?

     – Etscharntmin Sep 20 '23 at 13:32
  • if you write them down if groups of 4 going down you will see the pattern more easily – Etscharntmin Sep 20 '23 at 13:34
  • i explained it more here: https://math.stackexchange.com/questions/4772511/possibly-repeating-pattern-in-prime-numbers – Etscharntmin Sep 20 '23 at 14:43