Revision History for A306074
(Underlined text is an addition;
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Showing entries 1-10
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#27 by Bruno Berselli at Thu Jun 28 05:57:31 EDT 2018
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#26 by Joerg Arndt at Thu Jun 28 05:29:09 EDT 2018
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#25 by Jianing Song at Wed Jun 27 19:05:34 EDT 2018
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#24 by Jianing Song at Wed Jun 27 19:02:26 EDT 2018
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Jianing Song, <a href="/A306074/b306074.txt">Table of n, a(n) for n = 1..544</a>
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approved
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#23 by N. J. A. Sloane at Wed Jun 27 09:13:39 EDT 2018
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#22 by N. J. A. Sloane at Wed Jun 27 09:13:35 EDT 2018
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Bases in which 5 is a reciprocal-length unique-period prime.
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| COMMENTS
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A prime p is called a reciprocalunique-lengthperiod uniqueprime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a reciprocalunique-lengthperiod uniqueprime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
For every odd prime p, p is a reciprocal-lengtha unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a reciprocalunique-lengthperiod uniqueprime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a reciprocalunique-lengthperiod uniqueprime, with ord(b,p) >= 3. For p = 5, the nontrivial bases are 2, 3, 7.
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1/5 has period length 4 in base 2. Note that 3 and 5 are the only prime factors of 2^4 - 1 = 15, but 1/3 has period length 2, so 5 is a reciprocalunique-lengthperiod uniqueprime in base 2.
1/5 has period length 4 in base 3. Note that 2 and 5 are the only prime factors of 3^4 - 1 = 80, but 1/2 has period length 1, so 5 is a reciprocalunique-lengthperiod uniqueprime in base 3.
1/5 has period length 4 in base 7. Note that 2, 3 and 5 are the only prime factors of 7^4 - 1 = 2400, but 1/2 and 1/3 both have period length 1, so 5 is a reciprocalunique-lengthperiod uniqueprime in base 7.
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Cf. A040017 (reciprocal-length unique-period primes in base 10), A144755 (reciprocal-length unique primes in base 2).
Bases in which p is a reciprocal-length unique-period prime: A000051 (p=2), A306073 (p=3), this sequence (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).
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proposed
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#21 by N. J. A. Sloane at Wed Jun 27 08:58:52 EDT 2018
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#20 by N. J. A. Sloane at Wed Jun 27 08:58:49 EDT 2018
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Bases in which 5 is a reciprocal-length unique prime.
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| COMMENTS
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A prime p is called reciprocal-length unique in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is reciprocal-length unique in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
For every odd prime p, p is a reciprocal-length unique prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is reciprocal-length unique, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also reciprocal-length unique, with ord(b,p) >= 3. For p = 5, the nontrivial bases are 2, 3, 7.
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| EXAMPLE
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1/5 has period length 4 in base 2. Note that 3 and 5 are the only prime factors of 2^4 - 1 = 15, but 1/3 has period length 2, so 5 is reciprocal-length unique in base 2.
1/5 has period length 4 in base 3. Note that 2 and 5 are the only prime factors of 3^4 - 1 = 80, but 1/2 has period length 1, so 5 is reciprocal-length unique in base 3.
1/5 has period length 4 in base 7. Note that 2, 3 and 5 are the only prime factors of 7^4 - 1 = 2400, but 1/2 and 1/3 both have period length 1, so 5 is reciprocal-length unique in base 7.
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| CROSSREFS
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Cf. A040017 (reciprocal-length unique primes in base 10), A144755 (reciprocal-length unique primes in base 2).
Bases in which p is a reciprocal-length unique prime: A000051 (p=2), A306073 (p=3), this sequence (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).
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proposed
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#19 by Jianing Song at Wed Jun 27 06:46:30 EDT 2018
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#18 by Jianing Song at Wed Jun 27 06:45:25 EDT 2018
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| CROSSREFS
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Cf. A000051 (basesBases in which 2p is a unique prime: A000051 (p=2), A306073 (bases in which p=3 is a unique), this primesequence (p=5), A306075 (bases in which p=7 is a unique prime), A306076 (bases in which p=11 is a unique prime), A306077 (bases in which p=13 is a unique prime).
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