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?