Ours go to 211: Euler pseudoprimes to 47 prime bases (from Carmichael numbers)

Abstract

In this paper we show that a certain subset of the Carmichael numbers contains good Euler pseudoprimes, composite numbers that for many bases survive the Solovay-Strassen primality test. We present a classification of Carmichael numbers, and use the knowledge gained from this to create a fast algorithm to compute new Euler pseudoprimes, by multiplying already found Euler pseudoprimes. We use this algorithm to find many Euler pseudoprimes that are pseudoprimes for several consecutive prime bases starting at 2, hence for all integer bases up to that number. The best Euler pseudoprime we find survives up to 211, i.e., survives the first 47 prime bases.

Figures (6)

• Figure 1: Size of $\mathrm{epsp}(a)$ on logarithmic $y$-axis relative to $a$.
• Figure 2: Number of elements of $\mathrm{epsp}2(a)$ after multiplying two elements of $\mathrm{epsp}(37)$
• Figure 3: Schematic way of how we create $\mathrm{epsp}2(a)$
• Figure 4: Schematic way of how we create $\mathrm{epsp}3_{[1,2]}(a)$
• Figure 5: Schematic way of how we create $\mathrm{epsp} 7_{[1,6[3[1,2]^2]]}(a)$.

Theorems & Definitions (37)

• Definition 1: Fermat pseudoprimes
• Definition 2: Carmichael number
• Theorem 1: Korselt's criterion
• Theorem 2: Euler's criterion
• Definition 3: Euler pseudoprime
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof