Abstract
We extend the known tables of pseudosquares and pseudocubes, discuss the implications of these new data on the conjectured distribution of pseudosquares and pseudocubes, and present the details of the algorithm used to do this work. Our algorithm is based on the space-saving wheel data structure combined with doubly-focused enumeration, run in parallel on a cluster supercomputer.
Supported by a grant from the Holcomb Awards Committe, and computing resources provided by the Frank Levinson Supercomputing Center at Butler University.
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Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. of Math. 160(2), 781–793 (2004), http://dx.doi.org/10.4007/annals.2004.160.781
Bernstein, D.J.: Doubly focused enumeration of locally square polynomial values. In: High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, pp. 69–76. Amer. Math. Soc., Providence (2004)
Bernstein, D.J.: Proving primality in essentially quartic random time. Math. Comp. 76(257), 389–403 (2007), http://dx.doi.org/10.1090/S0025-5718-06-01786-8 (electronic)
Berrizbeitia, P., Müller, S., Williams, H.C.: Pseudocubes and primality testing. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 102–116. Springer, Heidelberg (2004)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)
Lukes, R.F., Patterson, C.D., Williams, H.C.: Some results on pseudosquares. Math. Comp. 65(213), 361–372, S25–S27 (1996)
Pomerance, C., Shparlinski, I.E.: On pseudosquares and pseudopowers. In: Combinatorial Number Theory, pp. 171–184. Walter de Gruyter, Berlin (2009)
Schinzel, A.: On pseudosquares. New Trends in Prob. and Stat. 4, 213–220 (1997)
Schroeppel, R.: Private communication (February 2010)
Sorenson, J.P.: The pseudosquares prime sieve. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 193–207. Springer, Heidelberg (2006)
Stephens, A.J., Williams, H.C.: An open architecture number sieve. In: Number theory and cryptography (Sydney, 1989). London Math. Soc. Lecture Note Ser., vol. 154, pp. 38–75. Cambridge Univ. Press, Cambridge (1990)
Williams, H.C.: Édouard Lucas and primality testing. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22. John Wiley & Sons Inc, New York (1998), A Wiley-Interscience Publication
Wooding, K.: The Sieve Problem in One- and Two-Dimensions. Ph.D. thesis, The University of Calgary, Calgary, AB (April 2010) http://math.ucalgary.ca/~hwilliam/files/wooding10thesis.pdf
Wooding, K., Williams, H.C.: Doubly-focused enumeration of pseudosquares and pseudocubes. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 208–221. Springer, Heidelberg (2006)
Wooding, K., Williams, H.C.: Improved primality proving with Eisenstein pseudocubes. In: Hanrot, G., Morain, F., Thomé, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 372–384. Springer, Heidelberg (2010)
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Sorenson, J.P. (2010). Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures. In: Hanrot, G., Morain, F., Thomé, E. (eds) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol 6197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14518-6_26
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DOI: https://doi.org/10.1007/978-3-642-14518-6_26
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