Powers of 2 and 5 without Zeroes

Theorem
The following $n \in \Z$ are such that both $2^n$ and $5^n$ have no zeroes in their decimal representation:
 * $0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$

Proof

 * {| border="1"

! align="center" style = "padding: 2px 10px" | $n$ ! align="center" style = "padding: 2px 10px" | $2^n$ ! align="center" style = "padding: 2px 10px" | $5^n$
 * align="right" style = "padding: 2px 10px" | $0$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $25$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $125$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $16$
 * align="right" style = "padding: 2px 10px" | $625$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $32$
 * align="right" style = "padding: 2px 10px" | $3125$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $64$
 * align="right" style = "padding: 2px 10px" | $15 \, 625$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $128$
 * align="right" style = "padding: 2px 10px" | $78 \, 125$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $512$
 * align="right" style = "padding: 2px 10px" | $1 \, 953 \, 125$
 * align="right" style = "padding: 2px 10px" | $18$
 * align="right" style = "padding: 2px 10px" | $262 \, 144$
 * align="right" style = "padding: 2px 10px" | $3 \, 814 \, 697 \, 265 \, 625$
 * align="right" style = "padding: 2px 10px" | $33$
 * align="right" style = "padding: 2px 10px" | $8 \, 589 \, 934 \, 592$
 * align="right" style = "padding: 2px 10px" | $116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
 * }
 * align="right" style = "padding: 2px 10px" | $512$
 * align="right" style = "padding: 2px 10px" | $1 \, 953 \, 125$
 * align="right" style = "padding: 2px 10px" | $18$
 * align="right" style = "padding: 2px 10px" | $262 \, 144$
 * align="right" style = "padding: 2px 10px" | $3 \, 814 \, 697 \, 265 \, 625$
 * align="right" style = "padding: 2px 10px" | $33$
 * align="right" style = "padding: 2px 10px" | $8 \, 589 \, 934 \, 592$
 * align="right" style = "padding: 2px 10px" | $116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
 * }
 * align="right" style = "padding: 2px 10px" | $116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
 * }

It is probable that $n = 33$ is the final instance.

Also see

 * Powers of 2 with no Zero in Decimal Representation
 * Powers of 5 with no Zero in Decimal Representation