arXiv 1001.41332010-01math.NT

双基表示长度的下界

Lower bounds on the lengths of double-base representations

Vassil S. Dimitrov, Everett W. Howe

备注：: 8 pages, LaTeX. Added DOIs for most references; corrected a minor error in arithmetic; made small copy-editing changes. To appear in Proc. Amer. Math. Soc
发表信息：: Proc. Amer. Math. Soc. 139 (2011), 3423-3430
DOI：: 10.1090/S0002-9939-2011-10764-0

摘要：

一个整数n的双基表示是一个表达式n = n_1 +. + n_r，其中n_i是（正或负）整数，除2或3外不能被素数整除;表示的长度是项数r。已知存在常数a > 0，使得每个整数n都有长度至多为log n / log log n的双基数表示。我们证明了存在一个常数c > 0，使得有无穷多个整数n的最短双基数表示的长度大于c log n /（log log n log log n）。我们的方法使我们能够找到最小的正整数，没有双基地表示的几个长度。特别地，我们证明了103是长度为2的最小无双基表示的正整数，4985是长度为3的最小无双基表示的正整数，641687是长度为4的最小无双基表示的正整数，并且326552783是长度为5的最小的没有双基数表示的正整数。

英文摘要：

A double-base representation of an integer n is an expression n = n_1 + ... + 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。