The largest prime

As $p$ is a prime then using FLT it can deduce

$2^{p+1} =2^p.2 \equiv 2.2 =4 \pmod{p}$

Similarly for other terms $3^{p+1} \equiv 9 \pmod{p} ; 5^{p+1} \equiv 25 \pmod{p} ; 7^{p+1} \equiv 49 \pmod{p}$

So, the remainder is $4+9+25+49=87=29 \times 3$

So, highest prime is $29$

Relevant wiki: Fermat's Little Theorem $2^{p+1}+3^{p+1}+5^{p+1}+7^{p+1}\equiv2^2+3^2+5^2+7^2\equiv87\equiv3\cdot29\equiv0\pmod{p}\implies p=\boxed{29}$ .