Intereting Posts

Positive integer solutions to $x^2+y^2+x+y+1=xyz$
Evaluating $\sum_{n \geq 1}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)$
Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.
Derivative of Riemann zeta, is this inequality true?
Isomorphism between quotient rings over finite fields
Mathematical concepts named after mathematicians that have become acceptable to spell in lowercase form (e.g. abelian)?
A nasty integral of a rational function
“Random” generation of rotation matrices
Does equicontinuity imply uniform continuity?
Prove that $a^2 + b^2 + c^2 $ is not a prime number
A Curious Binomial Sum Identity without Calculus of Finite Differences
If $1\leq p < \infty$ then show that $L^p()$ and $\ell_p$ are not topologically isomorphic
Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura
$(W_1+W_2+\cdots+W_n)^a \leq W_1^a +\cdots + W_n^a$ for $n$ integer, $n\geq 2$, $W\gt 0$ and $a$ constant, real, $0\lt a\lt 1$
How do you construct a function that is continuous over $(0,1)$ whose image is the entire real line?

Let $m>1$ be a natural number with $m \not\equiv 0 \pmod{10}$

Consider the powers $m^n$ , for which there is at least one digit not occurring

in the decimal representation.

Is there a largest $n$ with the desired property for any $m$? If so, define $n(m)$ to be this number.

- Starting digits of $2^n$.
- How to find last two digits of $2^{2016}$
- The last two digits of $9^{9^9}$
- How can the decimal expansion of this rational number not be periodic?
- Length of period of decimal expansion of a fraction
- Arrow notation and decimals.

Examples :

$$m=2 \rightarrow 2^{168} = 374144419156711147060143317175368453031918731001856$$

does not contain the digit $2$.

All the powers above up to $2^{10000}$ conatin all the digits, so

$2^{168}$ seems to be the biggest power with the desired property.

$$m=3 \rightarrow 3^{106} = 375710212613636260325580163599137907799836383538729$$

does not contain the digit $4$.

All the powers above up to $3^{10000}$ contain all the digits, so

$3^{106}$ seems to be the biggest power with the desired property.

So, probably $n(2)=168$ and $n(3)=106$ hold.

Is $n(m)$ defined for any $m$, and if yes, can reasonably sharp bounds be given?

- The binomial formula and the value of 0^0
- What better way to check if a number is a perfect power?
- A question comparing $\pi^e$ to $e^\pi$
- A 10-digit number whose $n$th digit gives the number of $(n-1)$s in it
- Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.
- How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$
- Finding the last two digits $123^{562}$
- Distinct digits in a combination of 6 digits
- How to use Fermat's little theorem to find $50^{50}\pmod{13}$?
- An “elementary” approach to complex exponents?

- Sum of $k$-th powers
- cup product in cohomology ring of a suspension
- Is an integer a sum of two rational squares iff it is a sum of two integer squares?
- Relationship Between Ratio Test and Power Series Radius of Convergence
- Prove that an equation has no elementary solution
- What is this pattern called?
- Connected open subsets in $\mathbb{R}^2$ are path connected.
- Karatsuba multiplication with integers of size 3
- Angle between two 3D vectors is not what I expected.
- Closed form solution of recurrence relation
- Is the Alexandroff double circle compact and Hausdorff?
- Equivalence of definitions of $C^k(\overline U)$
- Pile of $2000$ cards
- Prove that $\cos(x)$ doesn't have a limit as $x$ approaches infinity.
- Solution of nonlinear ODE: $x= yy'-(y')^2$