# A non-composite sequences

Can you provide a counterexample for a claim given below?

Inspired by Puzzle 937 I have formulated the following claim:

For any $$n > 0$$ let $$B = p_1 \cdot p_2 \cdot .... \cdot p_n$$ be the product of the first $$n$$ primes. Let $$X$$ be the smallest number, bigger than $$B^k/p_{n+1}$$ and coprime to $$B^k$$, where $$k$$ is a fixed positive integer. Define the number $$m_n$$ as $$X \cdot p_{n+1}-B^k$$ , then $$m_n$$ is either $$1$$ or prime.

Try it for yourself!

I was searching for counterexample using the following PARI/GP code:

CE(lb,ub,k)={
for(n=lb,ub,
B=prod(i=1,n,prime(i));
X=ceil((B^k)/prime(n+1));
while(gcd(X,B^k)!=1,
X++);
m=X*prime(n+1)-B^k;
if(!(ispseudoprime(m) || m==1),print(m)))
}

• @babanaCats, for $B^1=30$, aren't both $p_{n+1}$ and $X$ equal? – Collag3n Mar 7 '19 at 9:30
• I found no counterexamples for $1\le n\le 1000$, $1\le k\le 30$. – rogerl Mar 12 '19 at 22:50
• @rogerl Thank you for your investigation. – Peđa Terzić Mar 13 '19 at 12:16

After sieving $$p_n$$, numbers coprime to the primoral $$p_n\#$$ are prime at least up to $$p_{n+1}^2$$ ($$1$$ and square excluded of course)

e.g. $$p_n=5$$, numbers coprime to $$p_n\#=30=B$$ (also coprime to $$B^k$$) are $$\{1,7,11,13,17,19,23,29,31,37,41,43,47,49,...\}$$ and are prime up to $$49$$.

These numbers are generally expressed in this form: $$p_n\#\cdot i+\{1,p_{n+1},...\}$$

e.g. for $$p_n=3$$ they are expressed as $$6i+\{1,5\}$$ e.g. for $$p_n=5$$ they are expressed as $$30i+\{1,7,11,13,17,19,23,29\}$$

Obviously, $$X$$ is coprime to $$B^k$$ (it was picked in the coprime list) and so is $$p_{n+1}$$ which means $$X\cdot p_{n+1}$$ is also coprime to $$B^k$$(=$$30i$$ in our example) and since $$X\cdot p_{n+1}=B^k+m_n$$, we know $$m_n$$ is in that coprime list.

$$m_n=p_{n+1}\cdot x$$ where $$x=X-\frac{B^k}{p_{n+1}}$$ is smaller than the maximum gap between coprimes in the list (since we pick the first smaller coprime greater than $$\frac{B^k}{p_{n+1}}$$). If this maximum gap is smaller than $$p_{n+1}$$, then $$m_n and is therefore prime or $$1$$.

I think it was shown that the maximum gap between numbers coprime to $$p_n$$ is smaller than $$p_{n+1}$$, but I'll have a look when I have a bit more time.

edit: well....it was not http://oeis.org/A048670

• undelete of a quick try. Too quick in fact, but a failure that might...perhaps....give some ideas. I won't have enought time to dig more. – Collag3n Mar 13 '19 at 19:26