Question Number 510 by 123456 last updated on 21/Jan/15
![proof or given a counter example: if p,q are prines with p>q, and ∃s prime such s∈(q,p) then p−q≤Σ_(r∈(q,p),r is prime) r](https://www.tinkutara.com/question/Q510.png)
$${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${if}\:{p},{q}\:{are}\:{prines}\:{with}\:{p}>{q},\:{and}\:\exists{s}\:{prime} \\ $$$${such}\:{s}\in\left({q},{p}\right)\:{then} \\ $$$${p}−{q}\leqslant\underset{{r}\in\left({q},{p}\right),{r}\:{is}\:{prime}} {\sum}{r} \\ $$$$ \\ $$
Commented by prakash jain last updated on 22/Jan/15
![Finding a counter example will need three consecutive primes q, s, p such that s=q+k p>2q+k](https://www.tinkutara.com/question/Q512.png)
$$\mathrm{Finding}\:\mathrm{a}\:\mathrm{counter}\:\mathrm{example}\:\mathrm{will} \\ $$$$\mathrm{need}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{primes} \\ $$$${q},\:{s},\:{p}\:\mathrm{such}\:\mathrm{that} \\ $$$${s}={q}+{k} \\ $$$${p}>\mathrm{2}{q}+{k} \\ $$$$ \\ $$