1 Let $a$, $b$, $c$, $a+b-c$, $b+c-a$, $c+a-b$, $a+b+c$ be 7 distinct prime numbers, and the sum of two of $a$, $b$, $c$ is 800. Let $d$ be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of $d$. (2001 China Mathematical Olympiad Problem)
1. Let's assume $a0$, so $c<a+$ $b<a+c<b+c$, but one of $a+b$, $a+c$, $b+c$ is 800, so $c<800$. Also, $799=17 \times 47$, $798$ are not prime numbers, so $c \leqslant 797, d=2c \leqslant 1594$. Let $c=797$, $a+b=800$, noting that $b<c=797$, the smallest prime solution for $(a, b)=(13,787)$ (since $795,793=13 \times 61,789=3 \times 263$ are not prime numbers), at this point, $a+b-c=3$, $a-b+c=23$, $-a+b+c=1571$, $a+b+c=1597$ are all prime numbers, hence the maximum possible value of $d$ is 1594.