Giuga Sequence
المؤلف:
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R.
المصدر:
"Giuga,s Conjecture on Primality." Amer. Math. Monthly 103
الجزء والصفحة:
...
27-10-2020
989
Giuga Sequence
A finite, increasing sequence of integers ![<span style=]()
{n_1,...,n_m}" src="https://mathworld.wolfram.com/images/equations/GiugaSequence/Inline1.gif" style="height:15px; width:68px" /> such that
A sequence is a Giuga sequence iff it satisfies
 |
for 
, ..., 
. There are no Giuga sequences of length 2, one of length 3 (![<span style=]()
{2,3,5}" src="https://mathworld.wolfram.com/images/equations/GiugaSequence/Inline4.gif" style="height:15px; width:47px" />), two of length 4 (![<span style=]()
{2,3,7,41}" src="https://mathworld.wolfram.com/images/equations/GiugaSequence/Inline5.gif" style="height:15px; width:69px" /> and ![<span style=]()
{2,3,11,13}" src="https://mathworld.wolfram.com/images/equations/GiugaSequence/Inline6.gif" style="height:15px; width:76px" />), 3 of length 5 (![<span style=]()
{2,3,7,43,1805}" src="https://mathworld.wolfram.com/images/equations/GiugaSequence/Inline7.gif" style="height:15px; width:105px" />, ![<span style=]()
{2,3,7,83,85}" src="https://mathworld.wolfram.com/images/equations/GiugaSequence/Inline8.gif" style="height:15px; width:91px" />, and ![<span style=]()
{2,3,11,17,59}" src="https://mathworld.wolfram.com/images/equations/GiugaSequence/Inline9.gif" style="height:15px; width:98px" />), 17 of length 6, 27 of length 7, and hundreds of length 8. There are infinitely many Giuga sequences. It is possible to generate longer Giuga sequences from shorter ones satisfying certain properties.
REFERENCES:
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly 103, 40-50, 1996.
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