Wikipedia:Reference desk/Archives/Mathematics/2024 September 13

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September 13

Prove or disprove: These numbers are composite for all n>=2 such that these numbers are positive

Prove or disprove: These numbers are composite for all n>=2 such that these numbers are positive

1. $5^{n}+788$
2. $5^{n}-92$
3. $8^{n}+1$
4. $8^{n}+27$
5. $8^{n}+47$
6. $8^{n}-27$
7. $9^{n}-4$
8. $9^{n}-16$
9. $11^{n}+430$
10. $11^{n}-184$
11. $11^{n}-324$
12. $12^{n}-25$
13. $13^{n}-50$
14. $13^{n}-216$
15. $14^{n}+11$
16. $14^{n}-9$
17. $14^{n}-11$
18. $16^{n}-9$
19. $17^{n}+32$
20. $23^{n}+82$
21. $23^{n}-112$
22. $24^{n}-49$
23. $27^{n}+8$
24. $27^{n}-8$
25. $29^{n}+4$
26. $29^{n}+10$
27. $29^{n}+26$
28. $29^{n}-4$
29. $29^{n}-26$
30. $32^{n}+1$
31. $32^{n}+23$
32. $32^{n}-23$
33. $32^{n}-29$
34. $33^{n}-16$
35. $34^{n}-29$
36. $36^{n}-25$
37. $38^{n}+1$
38. $38^{n}+25$
39. $38^{n}+31$
40. $38^{n}-13$
41. $38^{n}-25$
42. $39^{n}-4$

118.170.47.16 (talk) 08:18, 13 September 2024 (UTC)

Re   #3: $8^{n}+1=(2^{n}+1)(4^{n}-2^{n}+1).$

Re   #4: $8^{n}+27=(2^{n}+3)(4^{n}-3\cdot 2^{n}+9).$

Re   #6: $8^{n}-27=(2^{n}-3)(4^{n}+3\cdot 2^{n}+9).$

Re   #7: $9^{n}-4=(3^{n}-2)(3^{n}+2).$

Re   #8: $9^{n}-16=(3^{n}-4)(3^{n}+4).$

Re #23: $27^{n}+8=(3^{n}+2)(9^{n}-2\cdot 3^{n}+4).$

Re #24: $27^{n}-8=(3^{n}-2)(9^{n}+2\cdot 3^{n}+4).$

Re #30: $32^{n}+1=(2^{n}+1)(16^{n}-8^{n}+4^{n}-2^{n}+1).$

Re #36: $36^{n}-25=(6^{n}-5)(6^{n}+5).$

These are all special cases of the difference of two nth powers.  --Lambiam 10:23, 13 September 2024 (UTC)

Do your own homework. SamuelRiv (talk) 15:34, 13 September 2024 (UTC)

Based on other posts with this type of number theoretic-focused content coming from IPs in the same geographic area (north Taiwan), I'm pretty sure this isn't meant to be homework. WP:CRUFT, perhaps, but this is the Reference Desk, and it seems to me that it's a lot less of an issue to have it here than elsewhere. GalacticShoe (talk) 16:52, 13 September 2024 (UTC)

For all of these which are not listed as always composite or having a prime, I tested $n\leq 10000$ and didn't find any primes.

1. Prime: $5^{10899}+788$ is probably prime. Note that $5^{n}+788$ is divisible by $3$ when $n$ is even, $13$ when $n\equiv 1\!\!{\pmod {4}}$, and $7$ when $n\equiv 5\!\!{\pmod {6}}$, so when there is such a prime then $n\equiv 3,7\!\!{\pmod {12}}$. Thanks to RDBury for helping establish compositeness originally for $n\leq 1000$.
2. Unknown: $5^{n}-92$ is divisible by $3$ when $n$ is odd, $13$ when $n\equiv 0\!\!{\pmod {4}}$, and $7$ when $n\equiv 0\!\!{\pmod {6}}$, so if there is such a prime then $n\equiv 2,10\!\!{\pmod {12}}$.
3. Always composite: $8^{n}+1$ can be factorized.
4. Always composite: $8^{n}+27$ can be factorized.
5. Always composite: $8^{n}+47$ is divisible by $3$ when $n$ is even, $5$ when $n\equiv 1\!\!{\pmod {4}}$, and $13$ when $n\equiv 3\!\!{\pmod {4}}$.
6. Always composite: $8^{n}-27$ can be factorized.
7. Always composite: $9^{n}-4$ can be factorized.
8. Always composite: $9^{n}-16$ can be factorized.
9. Unknown: $11^{n}+430$ is divisible by $3$ when $n$ is odd, $7$ when $n\equiv 1\!\!{\pmod {3}}$, $19$ when $n\equiv 2\!\!{\pmod {3}}$, and $13$ when $n\equiv 6\!\!{\pmod {12}}$, so if there is such a prime then $n\equiv 0\!\!{\pmod {12}}$.
10. Unknown: $11^{n}-184$ is divisible by $3$ when $n$ is even, $7$ when $n\equiv 2\!\!{\pmod {3}}$, $37$ when $n\equiv 3\!\!{\pmod {6}}$, and $13$ when $n\equiv 7\!\!{\pmod {12}}$, so if there is such a prime then $n\equiv 1\!\!{\pmod {12}}$.
11. Unknown: $11^{n}-324$ is divisible by $19$ when $n\equiv 0\!\!{\pmod {3}}$ and $7$ when $n\equiv 2\!\!{\pmod {3}}$, and it becomes a difference of squares if $n$ is even, so if there is such a prime then $n\equiv 1\!\!{\pmod {6}}$.
12. Always composite: $12^{n}-25$ is divisible by $13$ when $n$ is odd, and it becomes a difference of squares if $n$ is even.
13. Unknown: $13^{n}-50$ is divisible by $7$ when $n$ is even, so if there is such a prime then $n$ is odd.
14. Prime: $13^{5702}-216$ is probably prime. Note that $13^{n}-216$ is divisible by $7$ when $n$ is odd and $5$ when $n\equiv 0\!\!{\pmod {4}}$, and it becomes a difference of cubes if $n\equiv 0\!\!{\pmod {3}}$, so when there is such a prime then $n\equiv 2,10\!\!{\pmod {12}}$.
15. Always composite: $14^{n}+11$ is divisible by $5$ when $n$ is odd and $3$ when $n$ is even.
16. Always composite: $14^{n}-9$ is divisible by $5$ when $n$ is odd, and it becomes a difference of squares if $n$ is even.
17. Always composite: $14^{n}-11$ is divisible by $3$ when $n$ is odd and $3$ when $5$ is even.
18. Always composite: $16^{n}-9$ can be factorized.
19. Prime: $17^{9021}+32$ is probably prime. Note that $17^{n}+32$ is divisible by $3$ when $n$ is even, $5$ when $n\equiv 3\!\!{\pmod {4}}$, $7$ when $n\equiv 1\!\!{\pmod {6}}$, $19$ when $n\equiv 5\!\!{\pmod {9}}$, $181$ when $n\equiv 17\!\!{\pmod {36}}$, and $37$ when $n\equiv 29\!\!{\pmod {36}}$, so when there is such a prime then $n\equiv 9\!\!{\pmod {12}}$.
20. Prime: $23^{1926}+82$ is probably prime. Note that $23^{n}+82$ is divisible by $3$ when $n$ is odd, $7$ when $n\equiv 1\!\!{\pmod {3}}$, and $13$ when $n\equiv 2\!\!{\pmod {6}}$, so when there is such a prime then $n\equiv 0\!\!{\pmod {6}}$.
21. Prime: $23^{3409}-112$ is probably prime. Note that $23^{n}-112$ is divisible by $3$ when $n$ is even, $5$ when $n\equiv 3\!\!{\pmod {4}}$, and $17$ when $n\equiv 13\!\!{\pmod {16}}$ so when there is such a prime then $n\equiv 1,5,9\!\!{\pmod {16}}$.
22. Always composite: $24^{n}-9$ is divisible by $5$ when $n$ is odd, and it becomes a difference of squares if $n$ is even.
23. Always composite: $27^{n}+8$ can be factorized.
24. Always composite: $27^{n}-8$ can be factorized.
25. Always composite: $29^{n}+4$ is divisible by $3$ when $n$ is odd and $5$ when $n$ is even.
26. Prime: $29^{8096}+10$ is probably prime. Note that $29^{n}+10$ is divisible by $3$ when $n$ is odd, $13$ when $n\equiv 1\!\!{\pmod {3}}$, and $37$ when $n\equiv 2\!\!{\pmod {12}}$, so when there is such a prime then $n\equiv 0,6,8\!\!{\pmod {12}}$.
27. Always composite: $29^{n}+26$ is divisible by $5$ when $n$ is odd and $3$ when $n$ is even.
28. Always composite: $29^{n}-4$ is divisible by $5$ when $n$ is odd, and it becomes a difference of squares if $n$ is even.
29. Always composite: $29^{n}-26$ is divisible by $3$ when $n$ is odd, and it becomes a difference of squares if $5$ is even.
30. Always composite: $32^{n}+1$ can be factorized.
31. Always composite: $32^{n}+23$ is divisible by $11$ when $n$ is odd and $3$ when $n$ is even.
32. Always composite: $32^{n}-23$ is divisible by $3$ when $n$ is odd and $11$ when $n$ is even.
33. Unknown: $32^{n}-29$ is divisible by $3$ when $n$ is odd, $7$ when $n\equiv 0\!\!{\pmod {3}}$, $5$ when $n\equiv 2\!\!{\pmod {4}}$, and $13$ when $n\equiv 8\!\!{\pmod {12}}$, so if there is such a prime then $n\equiv 4\!\!{\pmod {12}}$.
34. Always composite: $33^{n}-16$ is divisible by $17$ when $n$ is odd, and it becomes a difference of squares if $n$ is even.
35. Always composite: $34^{n}-29$ is divisible by $5$ when $n$ is odd and $7$ when $n$ is even.
36. Always composite: $36^{n}-25$ can be factorized.
37. Unknown: $38^{n}+1$ is divisible by $3,13$ when $n$ is odd, $5,17$ when $n\equiv 2\!\!{\pmod {4}}$, and $41$ when $n\equiv 4\!\!{\pmod {8}}$, so if there is such a prime then $n\equiv 0\!\!{\pmod {8}}$.
38. Always composite: $38^{n}+25$ is divisible by $3$ when $n$ is odd and $13$ when $n$ is even.
39. Unknown: $38^{n}+31$ is divisible by $3$ when $n$ is odd, $5$ when $n\equiv 2\!\!{\pmod {4}}$, and $7$ when $n\equiv 4\!\!{\pmod {6}}$, so if there is such a prime then $n\equiv 0,8\!\!{\pmod {12}}$.
40. Always composite: $38^{n}-13$ is divisible by $3$ when $n$ is even, $5$ when $n\equiv 1\!\!{\pmod {4}}$, and $17$ when $n\equiv 3\!\!{\pmod {4}}$.
41. Always composite: $38^{n}-25$ is divisible by $13$ when $n$ is odd, and it becomes a difference of squares if $n$ is even.
42. Always composite: $39^{n}-4$ is divisible by $5$ when $n$ is odd, and it becomes a difference of squares if $n$ is even.

GalacticShoe (talk) 08:10, 15 September 2024 (UTC)

I'm guessing this is about the best one can do without actually discovering that some of the values are prime. I did some number crunching on the first sequence 5ⁿ+788 with n<1000. All have factors less than 10000 except for n = 87, 111, 147, 207, 231, 319, 351, 387, 471, 487, 499, 531, 547, 567, 591, 639, 687, 831, 919, 979. You can add more test primes to the list, for example if n%30 = 1 then 5ⁿ+788 is a multiple of 61, but nothing seems to eliminate all possible n. Wolfram Alpha says the smallest factor of 5⁸⁷+788 is 1231241858423, so it's probably not feasible to carry on without something more sophisticated than trial division. --RDBury (talk) 19:12, 15 September 2024 (UTC)

Thanks, RDBury. I've been using Alpertron, it's good at rapidly factorizing or finding low prime factors. GalacticShoe (talk) 22:20, 15 September 2024 (UTC)

Yes, Alpertron is very good and you can give it a file to factor, or specify a formula to factor. Bubba73 ^{You talkin' to me?} 05:44, 22 September 2024 (UTC)

Lines carrying rays

Not quite sure where to ask this but I decided to put it here. I apologize if this doesn't belong here.

I was recently reading about hyperbolic geometry and when reading the article Limiting parallel, I came across the statement "Distinct lines carrying limiting parallel rays do not meet." What exactly does it mean for a line to carry a ray? Is this standard mathematical terminology? TypoEater (talk) 18:14, 13 September 2024 (UTC)

Yes, this is the place for such a question, though you might also complain at Talk:Limiting parallel that the phrase is unclear. —Tamfang (talk) 22:51, 13 September 2024 (UTC)

I find the whole article unclear and confusing. Is it me?  --Lambiam 23:54, 13 September 2024 (UTC)