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Somer–Lucas pseudoprime









Somer–Lucas pseudoprime


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In mathematics, in particular number theory, an odd composite number N is a Somer–Lucas d-pseudoprime (with given d ≥ 1) if there exists a nondegenerate Lucas sequence U(P,Q){displaystyle U(P,Q)}U(P,Q) with the discriminant D=P2−4Q,{displaystyle D=P^{2}-4Q,}D=P^2-4Q, such that gcd(N,D)=1{displaystyle gcd(N,D)=1}gcd(N,D)=1 and the rank appearance of N in the sequence U(P, Q) is


1d(N−(DN)),{displaystyle {frac {1}{d}}left(N-left({frac {D}{N}}right)right),}frac{1}{d}left(N-left(frac{D}{N}right)right),

where (DN){displaystyle left({frac {D}{N}}right)}left({frac {D}{N}}right) is the Jacobi symbol.



Applications[edit]


Unlike the standard Lucas pseudoprimes, there is no known efficient primality test using the Lucas d-pseudoprimes. Hence they are not generally used for computation.



See also[edit]


Lawrence Somer, in his 1985 thesis, also defined the Somer d-pseudoprimes. They are described in brief on page 117 of Ribenbaum 1996.



References[edit]




  • Somer, Lawrence (1998). Bergum, Gerald E.; Philippou, Andreas N.; Horadam, A. F., eds. "On Lucas d-Pseudoprimes". Applications of Fibonacci Numbers. Springer Netherlands. 7: 369–375. doi:10.1007/978-94-011-5020-0_41..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Carlip, Walter; Somer, Lawrence (2007). "Square-free Lucas d-pseudoprimes and Carmichael-Lucas numbers". Czechoslovak Mathematical Journal. 57 (1).

  • Weisstein, Eric W. "Somer–Lucas Pseudoprime". MathWorld.


  • Ribenboim, P. (1996). "§2.X.D Somer-Lucas Pseudoprimes". The New Book of Prime Number Records (3rd ed.). New York: Springer-Verlag. pp. 131–132.











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