Euler-Jacobi Pseudoprime
المؤلف:
Guy, R. K.
المصدر:
"Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes." §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag
الجزء والصفحة:
...
23-1-2021
937
Euler-Jacobi Pseudoprime
An Euler-Jacobi pseudoprime to a base 
is an odd composite number 
such that 
and the Jacobi symbol 
satisfies
(Guy 1994; but note that Guy calls these simply "Euler pseudoprimes"). No odd composite number is an Euler-Jacobi pseudoprime for all bases 
relatively prime to it. This class includes some Carmichael numbers, all strong pseudoprimes to base 
, and all Euler pseudoprimes to base 
. An Euler pseudoprime is pseudoprime to at most 1/2 of all possible bases less than itself.
The first few base-2 Euler-Jacobi pseudoprimes are 561, 1105, 1729, 1905, 2047, 2465, ... (OEIS A047713), and the first few base-3 Euler-Jacobi pseudoprimes are 121, 703, 1729, 1891, 2821, 3281, 7381, ... (OEIS A048950). The number of base-2 Euler-Jacobi primes less than 
, 
, ... are 0, 1, 12, 36, 114, ... (OEIS A055551).
REFERENCES:
Guy, R. K. "Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes." §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.
Pinch, R. G. E. "The Pseudoprimes Up to 
." ftp://ftp.dpmms.cam.ac.uk/pub/PSP/.
Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, 1994.
Sloane, N. J. A. Sequences A047713/M5461, A048950, and A055551 in "The On-Line Encyclopedia of Integer Sequences."
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