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Consider all the natural numbers up to 1000 ending in the number n, which has a value in the set (1, 3, 7, 9). Calculate n^2-2 for each individual value of n up to 1000 and check if the result is prime. Save the set of these prime results for n=1, then another set for n=3, another set for n=7, and finally a set for n=9. Compare these sets quantitatively to determine how many primes they contain and list the results.

Up to 1000: Number of primes for n = 1: 30 Number of primes for n = 3: 32 Number of primes for n = 7: 33 Number of primes for n = 9: 27

Up to 10000: Number of primes for n = 1: 228 Number of primes for n = 3: 230 Number of primes for n = 7: 237 Number of primes for n = 9: 228

Up to 100000: Number of primes for n = 1: 1797 Number of primes for n = 3: 1837 Number of primes for n = 7: 1731 Number of primes for n = 9: 1774

Up to 1,000,000: Number of primes for n = 1: 14452 Number of primes for n = 3: 14592 Number of primes for n = 7: 14139 Number of primes for n = 9: 14261

Up to 10,000,000: Number of primes for n = 1: 1141550 Number of primes for n = 3: 1145834 Number of primes for n = 7: 1124059 Number of primes for n = 9: 1129097

The results continue to show a general trend of similar growth among prime sets for different values of n. However, the differences between the sets further increase as the range of values expands.

Betho's
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    It's known that before you square and subtract $2$ there are approximately the same number of primes of the four types. It's likely that there will be about the same number of primes of each type when you look at $n^2 -2$. – Ethan Bolker Jun 09 '23 at 00:59

1 Answers1

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It is not known that there are infinitely many primes of the form $n^2-2$, so no one is going to be able to prove anything about patterns in those primes.

On the other hand, no one doubts that there are infinitely many primes of the form $n^2-2$, and there are asymptotic estimates for the number of such primes up to $N$, and the number in any given congruence class (which covers the number ending in any given digit, since that comes down to congruence classes modulo ten). All such estimates have an error term which goes to infinity (which is why you are finding the differences increasing) but is negligible compared to the main term (so if you took ratios instead of differences you'd likely find the ratios converging to one).

There must be a typo in one of your results: up to $1,000,000$, $n=9$, can't be $1426$, that's much too small. Perhaps a digit got left off.

Gerry Myerson
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  • I adjusted the number to n=9 which was wrong. – Betho's Jun 09 '23 at 13:03
  • Compare the growth function of these quantities for each set of numbers n... – Betho's Jun 09 '23 at 17:15
  • It seems you have retracted your acceptance of this answer. It's your right to do that of course, but can you let me know whether there is something more that you need in an answer, Betho's? – Gerry Myerson Oct 19 '23 at 05:48
  • The number of primes for each value of n remains relatively close to each other, and the sets grow at similar rates as the range of natural numbers increases. This suggests that the property of being prime or not is somewhat evenly distributed among the four sets. – Betho's Oct 19 '23 at 10:49
  • Gauss's Theorem, refined by other mathematicians, establishes that prime numbers become more spaced out as natural numbers increase, but this is not the case here. – Betho's Oct 19 '23 at 10:54
  • The argument suggests that no one can prove anything about patterns in primes of the form n^2 - 2 because it's not known whether there are infinitely many such primes. However, this perspective overlooks important points: – Betho's Oct 19 '23 at 11:15
  • Observational Patterns: While we can't definitively prove the infinitude of primes of the form n^2 - 2, the data analysis discussed earlier reveals clear patterns in these primes, including differences in their distribution based on the value of n. – Betho's Oct 19 '23 at 11:15
  • Asymptotic Estimates: Asymptotic estimates offer valuable insights into the behavior of such primes, even if exact proofs are lacking. They help us understand the patterns and distribution of primes of the form n^2 - 2 up to a certain limit. – Betho's Oct 19 '23 at 11:15
  • Potential Patterns Exist: The observed trends in the data, such as differences between n values and the tendency for certain n values to generate more prime numbers, suggest that patterns do exist, even if they are not yet rigorously proven. – Betho's Oct 19 '23 at 11:15
  • In conclusion, while proving the infinitude of such primes remains an open question, patterns and insights can be gained from data analysis and asymptotic estimates, making it incorrect to claim that nothing can be said about patterns in primes of the form n^2 - 2. – Betho's Oct 19 '23 at 11:16