On the smallest primitive root of a safe prime

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

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A new prime $p$ for which the least primitive root $({\rm mod} p)$ and the least primitive root $({\rm mod} p^2)$ are not equal

Mathematics of Computation ◽

10.1090/s0025-5718-08-02090-5 ◽

2008 ◽

Vol 78 (266) ◽

pp. 1193-1195 ◽

Cited By ~ 2

Author(s):

A. Paszkiewicz

Keyword(s):

Primitive Root ◽

Mod P

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Explicit upper bound on the least primitive root modulo p2

International Journal of Number Theory ◽

10.1142/s1793042122500233 ◽

2021 ◽

pp. 1-7

Author(s):

Bo Chen

Keyword(s):

On Some Moves on Links and the Hopf Crossing Number

Mediterranean Journal of Mathematics ◽

10.1007/s00009-020-01611-6 ◽

2020 ◽

Vol 18 (1) ◽

Author(s):

Maciej Mroczkowski

Keyword(s):

Primitive Root ◽

Jones Polynomial ◽

Crossing Number ◽

Equivalence Classes ◽

Lens Spaces ◽

Root Of Unity ◽

Arrow Diagrams

AbstractWe consider arrow diagrams of links in $$S^3$$ S 3 and define k-moves on such diagrams, for any $$k\in \mathbb {N}$$ k ∈ N . We study the equivalence classes of links in $$S^3$$ S 3 up to k-moves. For $$k=2$$ k = 2 , we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a k-th primitive root of unity is unchanged by a k-move, when k is odd. It is multiplied by $$-1$$ - 1 , when k is even. It follows that, for any $$k\ge 5$$ k ≥ 5 , there are infinitely many classes of knots modulo k-moves. We use these results to study the Hopf crossing number. In particular, we show that it is unbounded for some families of knots. We also interpret k-moves as some identifications between links in different lens spaces $$L_{p,1}$$ L p , 1 .

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On Cyclotomic Numbers of Order Sixteen

Canadian Journal of Mathematics ◽

10.4153/cjm-1954-046-1 ◽

1954 ◽

Vol 6 ◽

pp. 449-454 ◽

Cited By ~ 2

Author(s):

Emma Lehmer

Keyword(s):

Linear Combination ◽

Primitive Root ◽

Number Of Solutions ◽

Integer Coefficients ◽

Cyclotomic Numbers ◽

Image Position ◽

Mod P

It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of (mod p),then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where,anda ≡ b ≡ 1 (mod 4),while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.

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