All Questions
9
questions
2
votes
1
answer
86
views
Show that for an odd integer $n ≥ 5$, $5^{n-1}\binom{n}{0}-5^{n-2}\binom{n}{1}+…+\binom{n}{n-1}$ is not a prime number.
I would prefer no total solutions and just a hint as to whether or not I’m at a dead end with my solution method.
So far this is my work:
From the binomial expansion,
$$\sum_{j=0}^n 5^{n-j}(-1)^j\...
- 183
4
votes
0
answers
243
views
Proper divisors of $P(x)$ congruent to 1 modulo $x$
Let $P(x) $ be a polynomial of degree $n\ge 4$ with integer coefficients and constant term equal to $1$. I am interested in Polynomials $P(x) $ such that for a fixed positive integer $b$, there are ...
- 234
1
vote
1
answer
170
views
Method to find if a polynomial has irrational roots?
Question
Let's say I define a polynomial $P(x)$ whose roots $\alpha_i$ and degree $> 1$.
We also add constraint that the coefficient of the highest power of $P(x)$ is $1$ and all other coefficients ...
- 1,177
3
votes
2
answers
286
views
When is $y=3x^2+3x+1$ a prime number in $\mathbb{Z}$ with $x \in \mathbb{Z}$?
The first few values of $y=3x^2+3x+1$ for integer values of $x$ are $7, 19, 37, 61, 91$, and $127$. I am wondering under what conditions of $x$ is $y$ a prime number?
I had initially hoped that ...
2
votes
0
answers
182
views
Why are polynomials of even powers better for Pollard's rho?
Taking all $C(900,2)$ combinations of the first 900 prime numbers, I defined $N = pq$, where $p$ and $q$ are a combination of primes. Then I factored $N$ using Pollard's Rho, counting how many ...
- 145
7
votes
1
answer
403
views
GCD of $n^a\,\prod\limits_{i=1}^k\,\left(n^{b_i}-n\right)$ for $n\in\mathbb{Z}$
Let $a$ be a nonnegative integer. For a given positive integer $k$, let $b_1,b_2,\ldots,b_k$ be odd integers greater than $1$. Using this result, it can be shown that, for each integer $n$, $$f_{a;...
- 49.8k
0
votes
1
answer
86
views
Can we efficiently decompose $n=ab$ given a "zero polynomial" modulo $n$?
Suppose that a polynomial $p=\sum_{0\leq j<n} u_jx^j$ with coefficients $0\leq u_j<n$ is constantly 0 modulo some $n$. If $n$ is a prime number, then $u=0$, because the Lagrange polynomials of ...
- 367
1
vote
0
answers
34
views
Suppose we have a polynomial over the integers with a non-zero leading coefficient over mod p.
Suppose we have a polynomial over the integers with a non-zero leading coefficient over $\mod p$.
Suppose $r$ is a zero of $f(r)$ is congruent to $0 \mod p$, show there exists polynomial $g(x)$ such ...
- 183
2
votes
0
answers
76
views
Polynomials producing Carmichael numbers
It is well known that $$(6n+1)(12n+1)(18n+1)$$ is a Carmichael number, if all factors are prime. I found the polynomials $$(6n+1)(12n+1)(18n+1)(36n+1)$$
$$(18n+1)(36n+1)(108n+1)(162n+1)$$ $$(20n+1)(...
- 85.1k