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A Proth number that is prime, i.e., a number of the form for odd , a positive integer, and . Factors of Fermat numbers are of this form as long as they satisfy the condition odd and . For example, the factor of is not a Proth prime since . (Otherwise, every odd prime would be a Proth prime.)
Proth primes satisfy Proth's theorem, i.e., a number of this form is prime iff there exists a number a such that is congruent to modulo . This provides an easy computational test for Proth primes. Yves Gallot has written a downloadable program for testing Proth primes and many of the largest currently known primes have been found with this program.
A Sierpiński number of the second kind is a number satisfying Sierpiński's composite number theorem, i.e., a Proth number such that is composite for every .
The first few Proth primes are 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, ... (OEIS A080076).
The following table gives the first few indices such that is prime for small .
OEIS | values of for which is prime | |
1 | 1, 2, 4, 8, 16, ... | |
3 | A002253 | 1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, ... |
5 | A002254 | 1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, ... |
7 | A032353 | 2, 4, 6, 14, 20, 26, 50, 52, 92, 120, ... |
9 | A002256 | 1, 2, 3, 6, 7, 11, 14, 17, 33, 42, 43, 63, ... |
REFERENCES:
Ballinger, R. "Proth Search Page." https://www.prothsearch.net/.
Caldwell, C. "Proth Prime." https://primes.utm.edu/glossary/page.php?sort=ProthPrime.
Caldwell, C. K. "Yves Gallot's Proth.exe: An implementation of Proth's Theorem for Windows." https://www.utm.edu/research/primes/programs/gallot/.
Keller, W. "The Least Prime of the Form for Certain Values of ." Abstr. Amer. Math. Soc. 9, 417-418, 1988.
McNamara, J. and Mills, M. "Factoring of Proth Numbers." https://www.fidn.org/proth1.html.
Sloane, N. J. A. Sequences A002253/M1318, A002254/M2635, A002256/M0751, A032353, and A080076 in "The On-Line Encyclopedia of Integer Sequences."
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مخاطر عدم علاج ارتفاع ضغط الدم
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اختراق جديد في علاج سرطان البروستات العدواني
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مدرسة دار العلم.. صرح علميّ متميز في كربلاء لنشر علوم أهل البيت (عليهم السلام)
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