The Strong Law of Small numbers

[sweety439]

1. Are the Fermat numbers 2^(2^n)+1 all primes?
2. Are the Mersenne numbers 2^p-1 all primes if p is prime?
3. Are the numbers 31, 331, 3331, 33331, 333331, ... all primes?
4. Are the numbers 3!-2!+1!, 4!-3!+2!-1!, 5!-4!+3!-2!+1!, 6!-5!+4!-3!+2!-1!, ... all primes?
5. Is n^2-79*n+1601 prime for all nonnegative integer n?
6. a_0 = 89, and for k > 0, a_k = 2*a_(k-1)+1, is a_n always prime?
7. If n is nonnegative integer, at least one of 6*n-1 and 6*n+1 must be prime?
8. If n > 12, can n+-1, n/2+-1, n/3+-1 be primes simultaneously?
9. Can 991*n^2+1 be square for n>=1?
10. Can three (positive or negative) cubes sum to 42?
11. Is 4*72^n-1 composite for all n>=1?
12. Is 8*48^n-1 composite for all n>=2?
13. Is 10^n+1 composite for all n>=3?
14. Is (18^p-1)/17 composite for all prime p>=3?
15. Does the sequence 1, 12, 123, 1234, ..., 123456789, 12345678910, 1234567891011, ... contain any primes?
16. Can tau(n) (where tau is Ramanujan's tau function) take a prime value?
17. Can Bell(n) be prime for n>55?
18. Can Fubini(n) be prime for n>13?
19. If 75353*2^n+1 is prime, must n be square?
20. Can p*2^p+1 be prime if p is prime?
21. Can n ellipses separate a plane to 2^n regions?
22. Draw n points on a circle, and write straight lines between any two of these points, the number of regions in this circle is 1, 2, 4, 8, 16 for n = 1, 2, 3, 4, 5, does the number of regions in this circle always 2^(n-1)?
23. a_0 = 1, and for k > 0, a_k = (1+a_0^2+a_1^2+...+a_(k−1)^2)/k, is a_n always integer?
24. Is ʃ(0,∞) (cos(x)cos(x/2)cos(x/3)...cos(x/n)) = π/2 for all positive integer n?
25. Are all finite orders of noncyclic simple groups divisible by 12?
26. If 2^(n-1) == 1 mod n, must n be prime?
27. If n divides Perrin(n), must n be prime?
28. If n is positive integer, can there be more primes of the form 4*m + 1 than of the form 4*m + 3, up to n?
29. If n is positive integer, can there be more primes of the form 3*m + 1 than of the form 3*m + 2, up to n?
30. If n is positive integer, can there be more primes of the form 12*m + 1 than of the form 12*m + r, for at least one of r = 5, 7, 11, up to n?
31. For all odd numbers n > 1, is there always a prime of the form n+2^r or n-2^r, for some positive integer r with 2^r < n?
32. Can all odd numbers > 1 be written as 2*m^2+p with p prime and m>=0?
33. Can all numbers > 24 be written as a sum of perfect power > 1 and a prime?
34. Can all even numbers > 2 be written as sum of two primes?
35. Can 24*n+2 be totient if n>=1?
36. Are all highly composite numbers also superabundant numbers?
37. Are all numbers coprime to 6 deficient numbers?
38. Are there any odd perfect numbers?
39. Are all Aliquot sequences terminate at 0, perfect number, amicable numbers, or sociable numbers?
40. Is there a number other than 10 such that sigma(n)/n = 9/5?
41. Is there any prime p such that 6^(p-1) == 1 mod p^2?
42. Is there any prime p such that F_(p-(p|5)) is divisible by p^2, where F is the Fibonacci number?
43. Is there any prime p such that binomial(2*p-1,p-1) == 1 mod p^4?
44. If p is prime, then 24^(p-1) == 1 mod p^2 if and only if 58^p-57^p is prime?
45. If n>1, then n^4+1 is prime if and only if 17*2^n-1 is prime?
46. Are all coefficients of all factors of x^n-1 always 0 and +-1?
47. Are all coefficients of the inverse of all factors of x^n-1 always 0 and +-1?
48. If Mp is Mersenne prime, must M(Mp) be Mersenne prime?
49. If Wp is Wagstaff prime, must W(Wp) be Wagstaff prime?
50. Can Mp and Wp be both prime for prime p > 127?
51. Can (Mp is prime) + (Wp is prime) + (p is of the form 2^n+-1 or 4^n+-3) be 2?
52. Is n^17+9 always coprime to (n+1)^17+9?
53. Is there an n such that eulerphi(x) = n has exactly one solution?
54. Is there a number whose 4th power can be written as three nonzero 4th powers?
55. Is there a number whose 5th power can be written as three nonzero 5th powers?
56. If L is the Liouville function, then is L(1) + L(2) + L(3) + ... + L(n) <= 0 for all n > 1?
57. If M is the Mertens function, then is M(n) <= 0 for all n > 1?
58. If M is the Mertens function, then is |M(n)| <= sqrt(n) for all n >= 1?
59. Is π(n) (the number of primes less than or equal to n) always < Li(n) (the Riemann's function)?
60. Do all complex numbers z such that zeta(z) = 0 has real part 1/2?

[LaurV]

When you say "The Strong Law of Small numbers" we expect you to have a counter-example to any of those. As you list them, some of them are unknown, and they may just become theorems soon, so no Guy's Law involved there. Like for example, you include Goldbach there, which has a very high probability to be true - as you go higher with n, there are more and more pairs of primes that add to n. So, this, probably, will never get a counterexample, to put it in Guy's basket...

[sweety439]

Originally Posted by LaurV

When you say "The Strong Law of Small numbers" we expect you to have a counter-example to any of those. As you list them, some of them are unknown, and they may just become theorems soon, so no Guy's Law involved there. Like for example, you include Goldbach there, which has a very high probability to be true - as you go higher with n, there are more and more pairs of primes that add to n. So, this, probably, will never get a counterexample, to put it in Guy's basket...

I know exactly which of these are unknown, and also know the smallest counterexample of others.

[science_man_88]

Originally Posted by sweety439

I know exactly which of these are unknown, and also know the smallest counterexample of others.

https://en.m.wikipedia.org/wiki/Stro..._small_numbers means you have counterexamples to them all.

[sweety439]

Originally Posted by science_man_88

https://en.m.wikipedia.org/wiki/Stro..._small_numbers means you have counterexamples to them all.

Some of them are completely created by me, such as 17, 18, 19, 33, 44, and the smallest counterexample of 17 is 2841, and the smallest counterexample of 33 is 1549