关于二的幂的三进制数的两个定理
On two conjectures concerning the ternary digits of powers of two
摘要：
埃尔德什指出，1、4和256是仅有的三进制表示仅由0和1组成的2的幂。Sloane解释说，除了$\{2^0,2^1,2^2,2^3,2^4,2^{15}\}$，每隔一个2的幂在其三进制表示中至少有一个0。本文给出的数值结果有力地支持了这些理论。特别地，我们用$n \leq 2 \cdot 3^{45} \approx 5.9 \times 10^{21}$验证了所有$2^n$的两种结构。我们的方法利用了一个简单的递归结构的数字$2^n$具有规定的模式，在其尾随的三进制数字。
英文摘要：
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.
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