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Old 2020-09-22, 20:52   #985
sweety439
 
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Nov 2016

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Riesel case: (k*b^n-1)/gcd(k-1,b-1)
Sierpinski case: (k*b^n+1)/gcd(k+1,b-1)

If k is not rational power of b, then:

* In Riesel case, (k*b^n-1)/gcd(k-1,b-1) has algebra factors if and only if k*b^n is perfect power (of the form m^r with r>1)
* In Sierpinski case, (k*b^n+1)/gcd(k+1,b-1) has algebra factors if and only if k*b^n is either perfect odd power (of the form m^r with odd r>1) or of the form 4*m^4

If k is rational power of b (let k = m^r, b = m^s):

* In Riesel case, (k*b^n-1)/gcd(k-1,b-1) has algebra factors if and only if n*s+r is composite
* In Sierpinski case, (k*b^n+1)/gcd(k+1,b-1) has algebra factors if and only if n*s+r is (not power of 2, if valuation(r,2) >= valuation(s,2)) (not of the form p*2^valuation(r,2) with p prime, if valuation(r,2) < valuation(s,2))

Last fiddled with by sweety439 on 2020-09-23 at 19:56
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