6
votes
3answers
134 views

Primes as a difference of powers

Find the smallest prime that cannot be written as $$|3^a - 2^b|$$ EDIT: I forgot to mention that $a$ and $b$ are whole numbers. I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but ...
1
vote
1answer
58 views

Does a solution exist where $p,q$ are odd primes and $p^a - q^b = p^c - q^d$ where $a > c > 1$ and $b > d > 1$

From my thinking so far, there is no solution. Is this an open question or is the answer well known? Here's my reasoning about this issue: If a solution exists, then: $$p^c(p^{a-c} - 1) = ...
0
votes
0answers
34 views

Find $a,b,c \ge 2$ and $p,q$ odd primes where $p^a - 1 = c*q^b$

I've been recently thinking about finding primes $p,q$ where the power of one divides the power of the other when subtracted by $1$. For example, if we remove the requirement that $p,q$ be odd ...
1
vote
1answer
33 views

Question on irreducible polynomials and primes.

Consider the polynomial $p(x) = 1+\sum_{i=1}^d a_i x^i$ where $a_i$ is binary and not all $a_i$ are $0$. Is it possible that $p(2^n)$ is prime for all integer $n>-1 ?$
11
votes
1answer
143 views

What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$

Mill's constant is a number such that $\lfloor A^{3^n}\rfloor$ is prime for all $n$. The existence of such an $A$ was proven in $1947$. I know little about number theory, but I am curious as to why ...
0
votes
1answer
277 views

How to check if Mersenne number is prime?

How can I prove that that Mersenne number, number of the type $2^n - 1$ is prime number. One theorem says that if $2^n - 1$ is prime than $n$ is prime number also. But this doesn't work vice versa. ...
5
votes
1answer
392 views

Prime decomposition of an integer: methods of determining the prime factors $ p_1, p_2, …, p_r$ and powers $k_1,k_2, …, k_r$

Any integer n can be written in the form $ n = p_1^{k_1}p_2^{k_2} ... p_r^{k_r} $, where the powers $ k_1, k_2, ...,k_r $ are integers and $ p_1, p_2, ..., p_r$ are primes. Now I am interested in ...
15
votes
2answers
349 views

Prove that for any nonnegative integer n the number $5^{5^{n+1}} + 5^{5 ^n} + 1$ is not prime

My math teacher gave us problems to work on proofs, but this problem has been driving me crazy. I tried to factor or find patterns in the numbers and all I can come up with is that for $n > 0$, the ...
2
votes
1answer
220 views

Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?

Let $x$ be a positive real number. Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$. Call this function $f(n,x)$. Can we give good upper and lower bounds of $f(n,x)$ ...
8
votes
1answer
454 views

Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?

Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$? $x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
2
votes
1answer
303 views