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Lower bounds on the lengths of double-base representations. (English) Zbl 1263.11086
Summary: A double-base representation of an integer $$n$$ is an expression $$n = n_1 + \cdots + n_r$$, where the $$n_i$$ are (positive or negative) integers that are divisible by no primes other than 2 or 3; the length of the representation is the number $$r$$ of terms. It is known that there is a constant $$a >0$$ such that every integer $$n$$ has a double-base representation of length at most $$a\log n / \log\log n$$. We show that there is a constant $$c>0$$ such that there are infinitely many integers $$n$$ whose shortest double-base representations have length greater than $$c\log n / (\log\log n \log\log\log n)$$.
Our methods allow us to find the smallest positive integers with no double-base representations of several lengths. In particular, we show that 103 is the smallest positive integer with no double-base representation of length 2, that 4985 is the smallest positive integer with no double-base representation of length 3, that 641687 is the smallest positive integer with no double-base representation of length 4, and that 326552783 is the smallest positive integer with no double-base representation of length 5.

##### MSC:
 11N56 Rate of growth of arithmetic functions 11A63 Radix representation; digital problems 11A67 Other number representations
Magma
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##### References:
 [1] Zs. Ádám, L. Hajdu, and F. Luca, Representing integers as linear combinations of \?-units, Acta Arith. 138 (2009), no. 2, 101 – 107. · Zbl 1230.11044 · doi:10.4064/aa138-2-1 · doi.org [2] Leonard M. Adleman, Carl Pomerance, and Robert S. Rumely, On distinguishing prime numbers from composite numbers, Ann. of Math. (2) 117 (1983), no. 1, 173 – 206. · Zbl 0526.10004 · doi:10.2307/2006975 · doi.org [3] Roberto Avanzi, Vassil Dimitrov, Christophe Doche, and Francesco Sica, Extending scalar multiplication using double bases, Advances in cryptology — ASIACRYPT 2006, Lecture Notes in Comput. Sci., vol. 4284, Springer, Berlin, 2006, pp. 130 – 144. · Zbl 1172.94558 · doi:10.1007/11935230_9 · doi.org [4] Valérie Berthé and Laurent Imbert, Diophantine approximation, Ostrowski numeration and the double-base number system, Discrete Math. Theor. Comput. Sci. 11 (2009), no. 1, 153-172. · Zbl 1221.11017 [5] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235 – 265. Computational algebra and number theory (London, 1993). · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125 · doi.org [6] V. S. Dimitrov, G. A. Jullien, and W. C. Miller, An algorithm for modular exponentiation, Inform. Process. Lett. 66 (1998), no. 3, 155 – 159. · Zbl 1078.94510 · doi:10.1016/S0020-0190(98)00044-1 · doi.org [7] Vassil Dimitrov, Laurent Imbert, and Pradeep K. Mishra, The double-base number system and its application to elliptic curve cryptography, Math. Comp. 77 (2008), no. 262, 1075 – 1104. · Zbl 1133.14034 [8] Vassil Dimitrov, Laurent Imbert, and Pradeep Kumar Mishra, Efficient and secure elliptic curve point multiplication using double-base chains, Advances in cryptology — ASIACRYPT 2005, Lecture Notes in Comput. Sci., vol. 3788, Springer, Berlin, 2005, pp. 59 – 78. · Zbl 1154.94388 · doi:10.1007/11593447_4 · doi.org [9] Christophe Doche and Laurent Imbert, Extended double-base number system with applications to elliptic curve cryptography, Progress in cryptology — INDOCRYPT 2006, Lecture Notes in Comput. Sci., vol. 4329, Springer, Berlin, 2006, pp. 335 – 348. · Zbl 1175.94076 · doi:10.1007/11941378_24 · doi.org [10] W. J. Ellison, On a theorem of S. Sivasankaranarayana Pillai, Séminaire de Théorie des Nombres, 1970 – 1971 (Univ. Bordeaux I, Talence), Exp. No. 12, Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1971, pp. 10. · Zbl 0276.10012 [11] Paul Erdős, Carl Pomerance, and Eric Schmutz, Carmichael’s lambda function, Acta Arith. 58 (1991), no. 4, 363 – 385. · Zbl 0734.11047 [12] Pradeep Kumar Mishra and Vassil Dimitrov, A combinatorial interpretation of double base number system and some consequences, Adv. Math. Commun. 2 (2008), no. 2, 159 – 173. · Zbl 1178.05008 · doi:10.3934/amc.2008.2.159 · doi.org [13] K. W. Wong, Edward C. W. Lee, L. M. Cheng, and Xiaofeng Liao, Fast elliptic scalar multiplication using new double-base chain and point halving, Appl. Math. Comput. 183 (2006), no. 2, 1000 – 1007. · Zbl 1126.94015 · doi:10.1016/j.amc.2006.05.111 · doi.org [14] ChangAn Zhao, FangGuo Zhang, and JiWu Huang, Efficient Tate pairing computation using double-base chains, Sci. China Ser. F 51 (2008), no. 8, 1096 – 1105. · Zbl 1210.94098 · doi:10.1007/s11432-008-0070-9 · doi.org
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