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Journal Article

On the Number of False Witnesses for a Composite Number

Paul Erdös and Carl Pomerance
Mathematics of Computation
Vol. 46, No. 173 (Jan., 1986), pp. 259-279
DOI: 10.2307/2008231
Stable URL: http://www.jstor.org/stable/2008231
Page Count: 21
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On the Number of False Witnesses for a Composite Number
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Abstract

If $a$ is not a multiple of $n$ and $a^{n - 1} \not\equiv 1 \operatorname{mod} n$, then $n$ must be composite and $a$ is called a "witness" for $n$. Let $F(n)$ denote the number of "false witnesses" for $n$, that is, the number of $a \operatorname{mod} n$ with $a^{n - 1} \equiv 1 \operatorname{mod} n$. Considered here is the normal and average size of $F(n)$ for $n$ composite. Also considered is the situation for the more stringent Euler and strong pseudoprime tests.

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