13
votes
3answers
553 views

Are there numbers that describe themselves in some base but not according to the pattern 6210001000?

Call the first digit of a number digit 0. The digit after that digit 1, and so on and so forth. In base 10, the number 6210001000 describes itself, because digit 0 is 6 and it has 6 0s. Digit 1 is 2 ...
5
votes
2answers
173 views

A question about decimal representation of irrational numbers.

Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the decimal representation of $\sqrt{2}$ ? Thanks !
1
vote
1answer
73 views

Prove that in any base the number of digits composing the repetitive mantissa of the reciprocal of a prime $p$ never exceeds $p-1$.

I was trying to find bases where the reciprocals of primes have a short repetitive mantissa. Here is what I found: http://imagizer.imageshack.us/a/img835/7738/c7gb.png The bases are on the left. The ...
0
votes
1answer
25 views

Binary expansion

I am trying to get my head around the left and right shift for binary expansion. The rules are: Shifting to the right ...
2
votes
1answer
34 views

Permuting digits in a power of $2$

Does there exist a natural number $N$ that is a power of $2$ whose digits (in the decimal representation) can be permuted to a different power of $2$? Thoughts: If such a number $N$ exists, then ...
2
votes
0answers
82 views

Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
1
vote
1answer
81 views

For which $n \in \mathbb{Z^+}$ is $(4n+9)/(2n^2+7n+6)$ a terminating decimal?

I saw this problem here. My approach: Let $A = (4n+9)/(2n^2+7n+6) = \frac{6}{2n + 3} - \frac{1}{n + 2}$ If $\frac{6}{2n + 3}$ and $\frac{1}{n + 2}$ are terminating decimals, then so is $A$. A ...
1
vote
0answers
108 views

Division of large numbers

Euclidean Algorithm Versus Horner Algorithm. I have come across this problem and I do not manage to find a good example for it. For all the numbers I pick there is no loss. Give an example of a ...
0
votes
1answer
41 views

Finding number of primes of the form $1010\ldots 101$ (in base 10) [closed]

How many primes among the positive integers, written as usual in the base $10$, are such that their digits are alternating $1$’s and $0$’s, beginning and ending with $1$?
1
vote
1answer
110 views

What are the last 20 digits of mega?

What are the last 20 digts of the number mega, which is "2 in a pentagon" in steinhaus-moser-notation ? In contrary to power towers or tetration, the ending digits are not stable. I found out that ...
2
votes
0answers
70 views

First digits of extremely large numbers (Generalization of “first digits of Graham's number”)

I found a question about the first digits of Graham's number and would like to generalize it : We want the first n digits of the number $a\uparrow^b c$. Which method is the most effective to do ...
8
votes
0answers
92 views

Biggest powers NOT containing all digits.

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 $n$ ...
5
votes
3answers
216 views

What is the smallest natural number n?

What is the smallest natural number n for which there is a natural k, such that, the lasts 2012 digit in the representation decimal of $n^k$ are equal to 1? I don't even know how to start with it ... ...
1
vote
0answers
75 views

Why are normal numbers important?

I'm studying normal numbers for my university dissertation - in particular whether or not algebraic numbers are normal. One thing I cannot find anywhere is why it matters. I'm not expecting a real ...
3
votes
1answer
86 views

Which Well-Known Numbers Are Alexandrian

A real number is said to be Decimal Alexandrian if its decimal representation contains every possible finite decimal sequence. It is a popular question whether $\pi$ is Decimal Alexandrian, or even in ...
1
vote
2answers
104 views

CS problem, turned to mathematics

I am trying to solve some of the projecteuler problems using a much of a programmers approach. However, I would like to get more into the math, and therefore would try to do some mathematical ...
3
votes
1answer
230 views

IBM Research Ponder This (June Challenge)

This was the challenge in last month's 'IBM Reseach Ponder This'. I just cannot get my head around the solution posted. Can someone explain further? Challenge Find a rational number (a fraction of ...
4
votes
1answer
145 views

Is every day tau day (or pi day) to some base?

People on my facebook wall are celebrating the fact that today is tau (2 pi) day—6/28. This got me thinking: today is only tau day because we represent numbers in decimal base. Could every day of the ...
8
votes
1answer
184 views

Last digits of a power of 2

Prove that there exists a power of 2 such that the last 1000 digits in its decimal representation are all 1 and 2. One fact that I think can be used in this problem: if $2^{n}=\cdots dn$ where ...
12
votes
2answers
240 views

Irrational numbers, decimal representation

Can this even be proved? (Or disproved?) Any irrational number without a 0 (zero) in its decimal representation is transcendental. Not sure where to start on this one...
1
vote
1answer
116 views

XOR for 10 and 20

I know that this is the XOR truth table. A B Q ------ 0 0 0 0 1 1 1 0 1 1 1 0 I have a = 10; and b=20; Their respective binaries are a=1010; and b=10100; a ...
1
vote
1answer
56 views

Explicit Fractional Basis Expansion

We can write any $x\in\mathbb R$ as its $b>1\in \mathbb N$ basis expansion: $$x=sgn(x)\sum_{d=-\infty}^\infty b^d \lfloor |x|b^{-d} - b\lfloor |x|b^{-d-1} \rfloor \rfloor$$ Can one come with the ...
11
votes
1answer
165 views

Square root of an integer has only even digits

Is there a non-square positive integer $n$, that $\sqrt{n}$ has only even digits in its decimal representation ?
3
votes
3answers
316 views

Characterising reals with terminating decimal expansions

Show that a number has a terminating decimal expansion if and only if, it is rational and when in lowest terms, its denominator is coprime to all primes other than $2$ and $5$. This is an unsolved ...
0
votes
1answer
215 views

Can the BBP formula be used to prove that Pi is normal?

Can the BBP formula be used to prove that Pi is normal? http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula http://en.wikipedia.org/wiki/Normal_number
9
votes
3answers
1k views

Cyclic numbers are characterized by the reciprocals of full reptend primes?

The number $142,857$ is widely known as a cyclic number, meaning consecutive multiples are cyclic permutations, i.e. $1 × 142,857 = 142,857$ $2 × 142,857 = 285,714$ $3 × 142,857 = ...
14
votes
2answers
519 views

Have all numbers with “sufficiently many zeros” been proven transcendental?

Any number less than 1 can be expressed in base g as $\sum _{k=1}^\infty {\frac {D_k}{g^k}}$, where $D_k$ is the value of the $k^{th}$ digit. If we were interested in only the non-zero digits of this ...
16
votes
2answers
1k views

Why is the decimal representation of 1/7 “cyclical”?

1/7 = 0.(142857)... with the digits in the parentheses repeating. I understand that the reason it's a repeating fraction is because 7 and 10 are coprime. But this...cyclical nature is something ...