Let $a, b, c, b+c-a, c+a-b, a+b-c$ and $a+b+c$ be $7$ distinct prime numbers. Among $a+b, b+c$ and $c+a$, only one of the three numbers is equal to $800$. If the difference of the largest prime number and the smallest prime number among the $7$ distinct prime numbers is described as $d$, then what is the highest possible value of $d$?
SOURCE: Bangladesh Math Olympiad
I didn't understand the pattern of rest prime numbers except $a, b$ and $c$ and their construction with the variable $a, b$ and $c$. Moreover, I couldn't catch out the probable number of the three numbers which were constructed by $a+b, b+c$ and $c+a$ whose real value is equal to $800$.
How can I get that $7$ prime numbers by applying any method or with that condition?
So, I really need some help and any reference to the post will be very helpful for my reaching to the conclusion. Thanks in advance.
Best Answer
Say $a+b=800$.
If $c\equiv 0\pmod 3$ then $c=3$ and then $a+b+c=803$ which is not a prime
If $c\equiv 1\pmod 3$ then $800+c\equiv 0\pmod 3$ so $800+c =3$ impossibile.
If $c\equiv -1\pmod 3$ then $800-c\equiv 0\pmod 3$ so $800-c =3\implies c=797$ so $a+b+c =1597$ and thus $\boxed{d= 1597-3 = 1594}$
Such a prime numbers exsist:
$a=787, b=13, c= 797,a+b+c=1597$
$a+b-c =3,\;\; a+c-b= 1571$ and $b+c-a = 23$