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Elliptic pseudoprime









Elliptic pseudoprime


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In number theory, a pseudoprime is called an elliptic pseudoprime for (E, P), where E is an elliptic curve defined over the field of rational numbers with complex multiplication by an order in Q(−d){displaystyle mathbb {Q} {big (}{sqrt {-d}}{big )}}mathbb{Q} big(sqrt{- d} big), having equation y2 = x3 + ax + b with a, b integers, P being a point on E and n a natural number such that the Jacobi symbol (−d | n) = −1, if (n + 1)P ≡ 0 (mod n).


The number of elliptic pseudoprimes less than X is bounded above, for large X, by


X/exp⁡((1/3)log⁡Xlog⁡log⁡log⁡X/log⁡log⁡X) .{displaystyle X/exp((1/3)log Xlog log log X/log log X) .} X / exp((1/3)log X logloglog X /loglog X) .


References[edit]



  • Gordon, Daniel M.; Pomerance, Carl (1991). "The distribution of Lucas and elliptic pseudoprimes". Mathematics of Computation. 57 (196): 825–838. doi:10.2307/2938720. Zbl 0774.11074..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}


External links[edit]


  • Weisstein, Eric W. "Elliptic Pseudoprime". MathWorld.












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