View Single Post 2021-01-09, 10:53 #116 sweety439   Nov 2016 22×3×5×47 Posts The simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b are of the form (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), this number has algebra factors if and only if: either * there is an integer r>1 such that both a*b^n and -c are perfect rth powers or * a*b^n*c is of the form 4*m^4 with integer m If (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebra factors, then it must be composite, the only exception is when it is either GFN (generalized Fermat number) base b or GRU (generalized repunit number) base b, in these two cases this number may be prime, the only condition is the n is power of 2 if it is GFN, and the n is prime if it is GRU GFNs and GRUs are the only simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b which are also cyclotomic numbers (i.e. numbers of the form Phi(n,b)/gcd(Phi(n,b),n), where Phi is cyclotomic polynomial) or Zsigmondy numbers Zs(n,b,1) (see Zsigmondy's theorem) GFNs and GRUs in bases 2<=b<=36: Code: base GFN family GRU family 2 1{0}1 {1} 3 {1}2 {1} 4 1{0}1 1{3}, {2}3 5 {2}3 {1} 6 1{0}1 {1} 7 {3}4 {1} 8 2{0}1, 4{0}1 1{7}, 3{7} 9 {4}5 1{4}, {6}7 10 1{0}1 {1} 11 {5}6 {1} 12 1{0}1 {1} 13 {6}7 {1} 14 1{0}1 {1} 15 {7}8 {1} 16 1{0}1 1{F}, 7{F}, {A}B, 2{A}B 17 {8}9 {1} 18 1{0}1 {1} 19 {9}A {1} 20 1{0}1 {1} 21 {A}B {1} 22 1{0}1 {1} 23 {B}C {1} 24 1{0}1 {1} 25 {C}D 1{6}, {K}L 26 1{0}1 {1} 27 1{D}E, 4{D}E 1{D}, 4{D} 28 1{0}1 {1} 29 {E}F {1} 30 1{0}1 {1} 31 {F}G {1} 32 2{0}1, 4{0}1, 8{0}1, G{0}1 1{V}, 3{V}, 7{V}, F{V} 33 {G}H {1} 34 1{0}1 {1} 35 {H}I {1} 36 1{0}1 1{7}, {U}V Note: we do not include the case where the "ground base" of the GFNs or GRUs is either perfect power or of the form -4*m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families for bases 2<=b<=36 are: Code: base GFN family GRU family 4 {1} 8 1{0}1 {1} 9 {1} 16 {1}, 1{5}, {C}D 25 {1} 27 {D}E {1} 32 1{0}1 {1} 36 {1} Note: the "ground base" of the GFNs or GRUs need not to be b (when b is perfect power), it may be root of b, it may also be negative integer which is root of b Last fiddled with by sweety439 on 2021-03-26 at 13:23  