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Old 2020-07-08, 18:05   #872
sweety439
 
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Nov 2016

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Originally Posted by sweety439 View Post
There are other k's excluded from the Riesel/Sierpinski problems (Riesel is still much more such k's)

* R30 k=1369:

for even n let n=2*q; factors to: (37*30^q - 1) * (37*30^q + 1)

odd n: covering set 7, 13, 19

* R88 k=400:

for even n let n=2*q; factors to: (20*88^q - 1) * (20*88^q + 1)

odd n: covering set 3, 7, 13

* R95 k=324:

for even n let n=2*q; factors to: (18*95^q - 1) * (18*95^q + 1)

odd n: covering set 7, 13, 229

* S55 k=2500:

odd n: factor of 7

n = = 2 mod 4: factor of 17

n = = 0 mod 4: let n=4q and let m=5*55^q; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S200 k=16:

odd n: factor of 3

n = = 0 mod 4: factor of 17

n = = 2 mod 4: let n = 4*q - 2 and let m = 20^q*10^(q-1); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

* S225 k=114244:

for even n let k=4*q^4 and let m=q*15^(n/2); factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

odd n: factor of 113

* R10 k=343:

n = = 1 mod 3: factor of 3

n = = 2 mod 3: factor of 37

n = = 0 mod 3: let n=3q and let m=7*10^q; factors to: (m - 1) * (m^2 + m + 1)

* R957 k=64:

n = = 1 mod 3: factor of 73

n = = 2 mod 3: factor of 19

n = = 0 mod 3: let n=3q and let m=4*957^q; factors to: (m - 1) * (m^2 + m + 1)

* S63 k=3511808:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3q and let m=152*63^q; factors to: (m + 1) * (m^2 - m + 1)

* S63 k=27000000:

n = = 1 mod 3: factor of 37

n = = 2 mod 3: factor of 109

n = = 0 mod 3: let n=3q and let m=300*63^q; factors to: (m + 1) * (m^2 - m + 1)

* R936 k=64:

n = = 0 mod 2: let n = 2q; factors to: (8*936^q - 1) * (8*936^q + 1)

n = = 0 mod 3: let n=3q; factors to: (4*936^q - 1) * [16*936^(2q) + 4*936^q + 1]

n = = 1 mod 6: factor of 37

n = = 5 mod 6: factor of 109
Of course,

* In the Riesel case if k and b are both r-th powers for an r>1

* In the Sierpinski case if k and b are both r-th powers for an odd r>1

* In the Sierpinski case if k is of the form 4*m^4, and b is 4th power

Then this k proven composite by full algebraic factors
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