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Old 2021-01-09, 10:53   #116
sweety439
 
Nov 2016

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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
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