Tagged Questions
8
votes
1answer
146 views
Sum of square root of primes
I was playing around with prime numbers and a question came into my mind:
Let $S(n)$ denote the sum of square roots of primes from $2$ to the $n$th prime number.
Are there infinitely many numbers $n$ ...
15
votes
5answers
784 views
Intervals that are free of primes
How can I prove that exists intervals as large as I want that are free of primes?
I mean, $\forall \ k \in \mathbb{N}, \exists \ k$ consecutive positive integers none of which is a prime.
2
votes
1answer
99 views
For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?
this is a question from a book I'm struggling with, please could you provide a clear proof
For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?
kind thanks
0
votes
1answer
55 views
For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically
For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
0
votes
0answers
94 views
Show that the field of p-adic numbers is complete
this is a question from a book I'm struggling with, please could you provide a clear proof
Show that the field of p-adic numbers is complete
i.e. that a sequence of p-adic numbers converges if and ...
0
votes
1answer
51 views
Polynomial that permutes residue classes
Prove that for any integers $d, e > 1$, the polynomial $f$ with integer coefficients permutes the residue classes modulo $p^d$ if and only if it permutes the residue classes modulo $p^e$ where $p$ ...
2
votes
1answer
421 views
Change of order of summation.
I feel like an idiot for asking this, so bear my stupidity.
I have the sum $\sum_{n\leq N} \sum_{p | n ; \ p \ prime} 1$, and I want to change the order of summation of these two sums I think it ...
9
votes
3answers
741 views
$\frac {n}{5} < \phi (n) < n$ for all $n > 1$?
Is it true that :
$\frac {n}{5} < \phi (n) < n$ for all $n > 1$
where $\phi (n)$ is Euler's totient function .
Since $\phi(n)$ has maximum value when $n$ is a prime it follows that ...
2
votes
1answer
124 views
Sum of exponents equivalent to card(composites less than or equal to x)?
Consider a cousin of the Chebyshev function:
$$t(x) = \sum \alpha_i $$ such that $$ p_i^{\alpha_i }= x, \ p_i \leq x$$
I speculated that $ t(x) \sim C(x)$, the composites $\leq(x)$
If $t(x) = ...
1
vote
3answers
169 views
How to prove $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$
$\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$, where $\tau(d)$ designates the number of positive divisors of d.
Now I only know that both sides are multiplicative functions, could ...
8
votes
4answers
168 views
Representing a number as a sum of at most $k$ squares
Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to
$$ n = x_1^2 + ...
120
votes
4answers
4k views
Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?
My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question:
I am thinking of a number...
It ...
