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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Some factors of the numbers $ G\sb{n}=6\sp{2n}+1$ and $ H\sb{n}=10\sp{2n}+1$


Author: Hans Riesel
Journal: Math. Comp. 23 (1969), 413-415
MSC: Primary 10.08
Corrigendum: Math. Comp. 24 (1970), 243.
Corrigendum: Math. Comp. 24 (1970), 243.
MathSciNet review: 0245507
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Abstract: All numbers $ {G_n} = {6^{2n}} + 1$ and $ {H_n} = {10^{2n}} + 1$ are searched for factors of the form $ p = u \cdot {2^8} + 1 < 3.88 \cdot {10^{11}}$ for $ s \geqq n + 1$, and odd $ u$. The search limit for $ u$ was 60000 for $ {G_n}$, and 156250 for $ {H_n}$. A number of factors are found in this range. The numbers $ {G_6}$ and $ {H_6}$, lacking small factors, are proved composite by calculating $ 5\mathop {^{(G - 1)/2}}\limits_6 (\bmod {G_6})$ and $ 3\mathop {^{(H - 1)/2}}\limits_6 (\bmod {H_6})$, the residues found being different from $ \pm 1$. The smallest numbers $ {G_n}$ and $ {H_n}$ with unknown characters are $ {G_{11}}$ and $ {H_{10}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1969-0245507-6
PII: S 0025-5718(1969)0245507-6
Article copyright: © Copyright 1969 American Mathematical Society