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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 Template:Div col

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. $39^n-4$

Template:Div col end 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 squares#Difference of two nth powers|difference of two Template:Mvarth 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 $n\le 10000$ 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 $n\equiv 1(\mathrm{mod}4)$, and $7$ when $n\equiv 5(\mathrm{mod}6)$, so when there is such a prime then $n\equiv 3,7(\mathrm{mod}12)$. Thanks to RDBury for helping establish compositeness originally for $n\le 1000$.
2. Unknown: $5^n-92$ is divisible by $3$ when $n$ is odd, $13$ when $n\equiv 0(\mathrm{mod}4)$, and $7$ when $n\equiv 0(\mathrm{mod}6)$, so if there is such a prime then $n\equiv 2,10(\mathrm{mod}12)$.
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 $n\equiv 1(\mathrm{mod}4)$, and $13$ when $n\equiv 3(\mathrm{mod}4)$.
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 $n\equiv 1(\mathrm{mod}3)$, $19$ when $n\equiv 2(\mathrm{mod}3)$, and $13$ when $n\equiv 6(\mathrm{mod}12)$, so if there is such a prime then $n\equiv 0(\mathrm{mod}12)$.
10. Unknown: $11^n-184$ is divisible by $3$ when $n$ is even, $7$ when $n\equiv 2(\mathrm{mod}3)$, $37$ when $n\equiv 3(\mathrm{mod}6)$, and $13$ when $n\equiv 7(\mathrm{mod}12)$, so if there is such a prime then $n\equiv 1(\mathrm{mod}12)$.
11. Unknown: $11^n-324$ is divisible by $19$ when $n\equiv 0(\mathrm{mod}3)$ and $7$ when $n\equiv 2(\mathrm{mod}3)$, and it becomes a difference of squares if $n$ is even, so if there is such a prime then $n\equiv 1(\mathrm{mod}6)$.
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 $n\equiv 0(\mathrm{mod}4)$, and it becomes a difference of cubes if $n\equiv 0(\mathrm{mod}3)$, so when there is such a prime then $n\equiv 2,10(\mathrm{mod}12)$.
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 $n\equiv 3(\mathrm{mod}4)$, $7$ when $n\equiv 1(\mathrm{mod}6)$, $19$ when $n\equiv 5(\mathrm{mod}9)$, $181$ when $n\equiv 17(\mathrm{mod}36)$, and $37$ when $n\equiv 29(\mathrm{mod}36)$, so when there is such a prime then $n\equiv 9(\mathrm{mod}12)$.
20. Prime: $23^{1926}+82$ is probably prime. Note that $23^n+82$ is divisible by $3$ when $n$ is odd, $7$ when $n\equiv 1(\mathrm{mod}3)$, and $13$ when $n\equiv 2(\mathrm{mod}6)$, so when there is such a prime then $n\equiv 0(\mathrm{mod}6)$.
21. Prime: $23^{3409}-112$ is probably prime. Note that $23^n-112$ is divisible by $3$ when $n$ is even, $5$ when $n\equiv 3(\mathrm{mod}4)$, and $17$ when $n\equiv 13(\mathrm{mod}16)$ so when there is such a prime then $n\equiv 1,5,9(\mathrm{mod}16)$.
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 $n\equiv 1(\mathrm{mod}3)$, and $37$ when $n\equiv 2(\mathrm{mod}12)$, so when there is such a prime then $n\equiv 0,6,8(\mathrm{mod}12)$.
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 $n\equiv 0(\mathrm{mod}3)$, $5$ when $n\equiv 2(\mathrm{mod}4)$, and $13$ when $n\equiv 8(\mathrm{mod}12)$, so if there is such a prime then $n\equiv 4(\mathrm{mod}12)$.
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 $n\equiv 2(\mathrm{mod}4)$, and $41$ when $n\equiv 4(\mathrm{mod}8)$, so if there is such a prime then $n\equiv 0(\mathrm{mod}8)$.
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 $n\equiv 2(\mathrm{mod}4)$, and $7$ when $n\equiv 4(\mathrm{mod}6)$, so if there is such a prime then $n\equiv 0,8(\mathrm{mod}12)$.
40. Always composite: $38^n-13$ is divisible by $3$ when $n$ is even, $5$ when $n\equiv 1(\mathrm{mod}4)$, and $17$ when $n\equiv 3(\mathrm{mod}4)$.
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)