Powers of 2 and 5 without Zeroes
Jump to navigation
Jump to search
Theorem
The following $n \in \Z$ are such that both $2^n$ and $5^n$ have no zeroes in their decimal representation:
- $0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$
This sequence is A007496 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
$n$ $2^n$ $5^n$ $0$ $1$ $1$ $1$ $2$ $5$ $2$ $4$ $25$ $3$ $8$ $125$ $4$ $16$ $625$ $5$ $32$ $3125$ $6$ $64$ $15 \, 625$ $7$ $128$ $78 \, 125$ $9$ $512$ $1 \, 953 \, 125$ $18$ $262 \, 144$ $3 \, 814 \, 697 \, 265 \, 625$ $33$ $8 \, 589 \, 934 \, 592$ $116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
It is probable that $n = 33$ is the final instance.
$\blacksquare$
Also see
- Powers of 2 with no Zero in Decimal Representation
- Powers of 5 with no Zero in Decimal Representation
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $33$