Questions tagged [decimal-expansion]
For questions about decimal expansion, both practical and theoretical.
1,287 questions
-4 votes
0 answers
84 views
Is there a largest power of $2$ that does not contain $67$ in its decimal expansion? [duplicate]
Phrased formally: Does there exist a natural number $N$ such that for all $n>N$, there is some natural number $k$ such that $67 \times 10^k \leq 2^n < 68 \times 10^k$? I conjecture yes for two ...
2 votes
0 answers
100 views
2 base 10 = 10 base 2 - how unique?
Surely I'm not the first to observe! But it's a bit of a cheat: In which base do we read the bases when stating $a_b=b_a$? (I write $a_b$ for $a$ base $b$.) Suggestion: Like in the title, in a "...
- 853
-1 votes
1 answer
97 views
How do we pick how many decimals to use when solving problems? [closed]
I was working on an exercise from a chemistry textbook which was about an application of the osmotic pressure formula. When I checked the solution the answer give was $62.05$ while I was getting as a ...
- 133
-1 votes
1 answer
62 views
Integers whose decimal representation is suffix of their binary representation
For any given t>1, I need to find all positive integers p whose decimal representation i) has length t ii) is a suffix (not prefix) of the binary representation of p. 'Representation' here means ...
3 votes
0 answers
75 views
Concatenations of permutations are distinct primes
The set of positive integers $\left\{ 343, 441, 603 \right\}$ has the property that concatenations of the $3! = 6$ permutations of decimal representations of those integers are decimal representations ...
- 1,507
0 votes
0 answers
64 views
Digit-cube alternative sum dynamic
Definition (The Rule): Let '$n$' be a positive integer with digits $$n = d_1 d_2 d_3 \dots d_k$$ Cube each digit. Assign alternating signs: $ d_1^3-d_2^3+d_3^3-d_4^3...d_k$ Conjecture: For every ...
-1 votes
2 answers
177 views
Prove $0.6\dot{1}\dot{2} = \frac{101}{165}$ [closed]
For those who wants a bit background, this is a proof question for international GCSE paper, for students aged 14-16. This is what the candidate has written. Obviously, it is NOT the usual method we ...
- 274
3 votes
3 answers
195 views
Proving a number of base 2 non-recurring if converted to base 10 having the same decimal point position from the right of the number
I have a precision $P_2$ and $P_{10}$ defined as the position of the decimal point from the right of the number. $P_2$ for base 2, $P_{10}$ for base 10, for example: $$ \begin{align*} \\ P_2(110....
- 133
-2 votes
1 answer
158 views
Just noticed this: numbers where digit sum equals digit product??
Okay so this might sound super basic but it kinda tripped me up. Was messing around with numbers and saw that 123 has this weird thing going on: Sum of digits = 1 + 2 + 3 = 6 Product of digits = 1 × 2 ...
1 vote
0 answers
176 views
Is $37$ the only prime number $p$ such that the repeating part of $\dfrac{1}{p}$ is one less than an even perfect number?
Is $37$ the only prime number $p$ such that the repeating part of $\dfrac{1}{p}$ is one less than an even perfect number? This question was asked in our school contest that ended a few months ago. I ...
- 1,020
4 votes
1 answer
826 views
Is the leading digit of the decimal expansion of the prime zeta function at $n$ equal to the first digit of $5^n$, for all integers $n \geq 10$?
A few days ago, I submitted to the OEIS the sequence of the leading digit (i.e., the most significant nonzero digit) of the decimal expansion of the prime zeta function at $n$. Surprisingly, another ...
- 1,374
0 votes
0 answers
72 views
What is the unit's digit of $a^b,$ where $a$ is a rational number and $b$ is an integer?
Given $a^b$, where $a$ is a rational number and $b$ is an integer, what is the value of the unit's digit? Question Origin After showing someone how to find the unit's digit of $a^b$ given $a$ and $b$ ...
- 105
0 votes
1 answer
75 views
Matching a number to the average of its digits: Can we count the base two solutions?
This question asks for nonzero numbers whose values match the average of their own digits. Such numbers are not immediatelly evident by intuition, but it is not difficult to show that there are in ...
- 51.2k
11 votes
5 answers
3k views
Do all irrational numbers contain each other? [closed]
From my understanding, both $e$ and $\pi$ are both non-terminating and non-repeating numbers that are infinitely long and they contain every possible sequence of digits. If that is true, does that ...
- 382
1 vote
0 answers
82 views
Bounds of a limit related with sum of digits
Background: Assume $b>1$ is a positive integer. For $n,m\in\mathbb{N}$ , let $S_b(n)$ denote the sum of digits of $n$ in base $b$, and $$F_b(m):=\displaystyle\sum_{S_b(n)=m}\frac{1}{n}$$ In this MO ...
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