Giuga Sequence

A finite, increasing sequence of integers {n_1,...,n_m} such that

 sum_(i=1)^m1/(n_i)-product_(i=1)^m1/(n_i) in N.

A sequence is a Giuga sequence iff it satisfies


for i=1, ..., m. There are no Giuga sequences of length 2, one of length 3 ({2,3,5}), two of length 4 ({2,3,7,41} and {2,3,11,13}), 3 of length 5 ({2,3,7,43,1805}, {2,3,7,83,85}, and {2,3,11,17,59}), 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.

See also

Carmichael Sequence

Explore with Wolfram|Alpha


Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly 103, 40-50, 1996.

Referenced on Wolfram|Alpha

Giuga Sequence

Cite this as:

Weisstein, Eric W. "Giuga Sequence." From MathWorld--A Wolfram Web Resource.

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