论文标题
关于两个权力的三元数字的两个猜想
On two conjectures concerning the ternary digits of powers of two
论文作者
论文摘要
Erdős认为1、4和256是两个三元表示仅由0和1组成的唯一功能。斯隆（Sloane）猜想，除了$ \ {2^0,2^1,2^1,2^2,2^3,2^4,2^{15} \} $之外，两个其他两个功率在其三元表示中至少具有一个0。在本文中，为这些猜想的强烈支持给出了数值结果。特别是，我们用$ n \ leq 2 \ cdot 3^{45} \大约5.9 \ times 10^{21} $验证所有$ 2^n $的两个猜想。我们的方法利用了一个简单的递归构造数字$ 2^n $，在其尾随三元数字中规定了图案。
Erdős conjectured that 1, 4, and 256 are the only powers of two whose ternary representations consist solely of 0s and 1s. Sloane conjectured that, except for $\{2^0,2^1,2^2,2^3,2^4,2^{15}\}$, every other power of two has at least one 0 in its ternary representation. In this paper, numerical results are given in strong support of these conjectures. In particular, we verify both conjectures for all $2^n$ with $n \leq 2 \cdot 3^{45} \approx 5.9 \times 10^{21}$. Our approach makes use of a simple recursive construction of numbers $2^n$ having prescribed patterns in their trailing ternary digits.
