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**1 - 1**of**1**### FINDING C3-STRONG PSEUDOPRIMES

, 2004

"... Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3) the signature, andH = h1h2h3 the height of N, respectively. We call N a C3-number if it is a Carmichael number with each prime factor qi≡3 m ..."

Abstract
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Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3) the signature, andH = h1h2h3 the height of N, respectively. We call N a C3-number if it is a Carmichael number with each prime factor qi≡3 mod 4. If N is a C3-number and a strong pseudoprime to the t bases bi for 1 ≤ i ≤ t, we call NaC3-spsp(b1,b2,...,bt). Since C3-numbers have probability of error 1/4 (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of ψm (the smallest strong pseudoprime to all the first m prime bases). If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. In this paper, we first describe an algorithm for finding C3-spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 C3-spsp(2)’s < 1024 with heights < 109. We then give an overview of the 21978 C3spsp(2)’s and tabulate 54 of them, which are C3-spsp’s to the first 8 prime bases up to 19; three numbers are spsp’s to the first 11 prime bases up to 31. No C3-spsp’s < 1024 to the first 12 prime bases with heights < 109 were found. We conjecture that there exist no C3-spsp’s < 1024 to the first 12 prime bases with heights ≥ 109 and so that