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Math 412: Number TheoryLecture 14 Order of an integer and Primitive root
Gexin [email protected]
College of William and Mary
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
The number 142857
142857
142857 ∗ 2 =285714
142857 ∗ 3 =428571
142857 ∗ 4 =571428
142857 ∗ 5 =714285
142857 ∗ 6 =857142
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Let ordna be the order of a modulo n.
Thm: (a, n) = 1, then ax ≡ 1 (mod n) if and only if ordna|x .
As a consequence, ordn(a)|φ(n).
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Let ordna be the order of a modulo n.
Thm: (a, n) = 1, then ax ≡ 1 (mod n) if and only if ordna|x .
As a consequence, ordn(a)|φ(n).
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Let ordna be the order of a modulo n.
Thm: (a, n) = 1, then ax ≡ 1 (mod n) if and only if ordna|x .
As a consequence, ordn(a)|φ(n).
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Thm: If (a, n) = 1, then ai ≡ aj (mod n) if and only if i ≡ j(mod ordna).
If (a, n) = 1 and a ≡ b (mod n), then ordna = ordnb.
If b = a−1 is the inverse of a modulo n, then ordna = ordna−1.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Thm: If (a, n) = 1, then ai ≡ aj (mod n) if and only if i ≡ j(mod ordna).
If (a, n) = 1 and a ≡ b (mod n), then ordna = ordnb.
If b = a−1 is the inverse of a modulo n, then ordna = ordna−1.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Thm: If (a, n) = 1, then ai ≡ aj (mod n) if and only if i ≡ j(mod ordna).
If (a, n) = 1 and a ≡ b (mod n), then ordna = ordnb.
If b = a−1 is the inverse of a modulo n, then ordna = ordna−1.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
If n|m, then ordna|ordma.
If (m1,m2) = 1, then ordm1m2a = [ordm1a, ordm2a].
If n =∏
i ptii and (a, n) = 1, then
ordna|[ordpt11 (a), φpt22 (a), . . . , φptkk (a)]
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
If n|m, then ordna|ordma.
If (m1,m2) = 1, then ordm1m2a = [ordm1a, ordm2a].
If n =∏
i ptii and (a, n) = 1, then
ordna|[ordpt11 (a), φpt22 (a), . . . , φptkk (a)]
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
If n|m, then ordna|ordma.
If (m1,m2) = 1, then ordm1m2a = [ordm1a, ordm2a].
If n =∏
i ptii and (a, n) = 1, then
ordna|[ordpt11 (a), φpt22 (a), . . . , φptkk (a)]
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Primitive root
Def: if ordna = φ(n), then a is a primitive root of (modulo) n.
Thm: if (r , n) = 1, and r is a primitive root of n, then r , r2, . . . , rφ(n)
form a reduced system of residues modulo n.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
Primitive root
Def: if ordna = φ(n), then a is a primitive root of (modulo) n.
Thm: if (r , n) = 1, and r is a primitive root of n, then r , r2, . . . , rφ(n)
form a reduced system of residues modulo n.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
If ordna = t, then ordn(au) = t
(t,u) .
If r is a primitive root of n, then ru is a primitive root of n if and onlyif (u, φ(n)) = 1.
Furthermore, if n has a primitive root, then n has φ(φ(n))incongruent primitive roots.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
If ordna = t, then ordn(au) = t
(t,u) .
If r is a primitive root of n, then ru is a primitive root of n if and onlyif (u, φ(n)) = 1.
Furthermore, if n has a primitive root, then n has φ(φ(n))incongruent primitive roots.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
If ordna = t, then ordn(au) = t
(t,u) .
If r is a primitive root of n, then ru is a primitive root of n if and onlyif (u, φ(n)) = 1.
Furthermore, if n has a primitive root, then n has φ(φ(n))incongruent primitive roots.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
142857 is a cyclic number
In general, if the digit period of 1/p (p prime) is p − 1, then thedigits represent a cyclic number.
In other words, if 10 is a primitive root for p, then 1/p has digitperiod p − 1, and the cyclic number has the form (10p−1 − 1)/p. Inthis case, p is called a long prime for 10.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
142857 is a cyclic number
In general, if the digit period of 1/p (p prime) is p − 1, then thedigits represent a cyclic number.
In other words, if 10 is a primitive root for p, then 1/p has digitperiod p − 1, and the cyclic number has the form (10p−1 − 1)/p. Inthis case, p is called a long prime for 10.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
142857 is a cyclic number
In general, if the digit period of 1/p (p prime) is p − 1, then thedigits represent a cyclic number.
In other words, if 10 is a primitive root for p, then 1/p has digitperiod p − 1, and the cyclic number has the form (10p−1 − 1)/p. Inthis case, p is called a long prime for 10.
Gexin Yu [email protected] Math 412: Number Theory Lecture 14 Order of an integer and Primitive root
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