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Search: id:A082654
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| A082654 |
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Order of 4 mod n-th prime: least k such that prime(n) divides 4^k-1, n>=2. |
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3
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| 0, 1, 2, 3, 5, 6, 4, 9, 11, 14, 5, 18, 10, 7, 23, 26, 29, 30, 33, 35, 9, 39, 41, 11, 24, 50, 51, 53, 18, 14, 7, 65, 34, 69, 74, 15, 26, 81, 83, 86, 89, 90, 95, 48, 98, 99, 105, 37, 113, 38, 29, 119, 12, 25, 8, 131, 1, 34, 135, 46, 35, 47, 146, 51, 155, 78, 158, 15, 21, 173, 174
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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The period of the expansion of 1/p, base N (where N=4), is equivalent to determining for base integer 4, the period of the sequence 1, 4, 4^2, 4^3...mod p. Thus the cycle length for base 4, 1/7 = .021021021...(cycle length 3).
The cycle length, base 4, mod p, is equivalent to "clock cycles", given angle A, then the algebraic identity for the doubling angle, 2A.
Examples: Given Cos A, f(x) for 2A = 2x^2 - 1, seed 2 Pi /7 i.e. (.623489801 == (arrow), -.222520934... == -.900968867...== .623489801...(cycle length 3). Given 2 Cos A, the algebraic identity for 2 Cos 2A, f(x) = x^2 - 2; e.g. Given seed 2 Cos A = 2 Pi /7, the 3 cycle is 1.246979604...== .445041867...== -1.801937736...== back to 1.24697... Likewise, the doubling function given Sin^2 A, f(x) for Sin^2 2A = 4x(1 - x), the logistic equation; getting cycle length of 3 using the seed Sin^2 2 Pi /7. Similarly, the doubling function for Tan 2A given Tan A, where A = 2 Pi /7 gives 2x/(1 - x^2), cycle length of 3. The doubling function for Cot 2A given Cot A, with A = 2 Pi /7 gives (x^2 - 1)/2x, cycle length of 3. Note that (x^2 - 1)/2x = Sinh Ln x; and is also generated from using Newton's method on x^2 + 1 = 0.
Consider the odd pseudoprimes, composite numbers x such that 2^(x-1) = 1 mod x, that have prime(n) as a factor. It appears that all such x can be factored as prime(n) * (2 a(n) k + 1) for some integer k. For example, the first few pseudoprimes having the factor 31 are 31*11, 31*91, 31*141, and 3*151. The 11th prime is 31 and a(11) = 5. Therefore all the cofactors of 31 should have the form 10k+1, which is clearly true. - T. D. Noe (noe(AT)sspectra.com), Jun 10 2003
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1964; Table 48, pages 98-99.
John H. Conway & R. K. Guy, The Book of Numbers, Springer-Verlag, 1996, pages 207-208, Periodic Points.
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FORMULA
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Least exponent k for which 4^k is congruent to 1 mod p.
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EXAMPLE
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4th prime is 7 and mod 7, 4^3 = 1, so a(4) = 3.
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MATHEMATICA
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Join[{0}, Table[MultiplicativeOrder[4, Prime[n]], {n, 2, 100}]]
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CROSSREFS
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Cf. A014664, A002326, A036116, A036117.
Sequence in context: A130386 A137760 A054077 this_sequence A072636 A001600 A000036
Adjacent sequences: A082651 A082652 A082653 this_sequence A082655 A082656 A082657
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2003
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EXTENSIONS
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More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 17 2003
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