Just for fun!-XII

d^4mod5=1 for gcd(d,5)=1 and 0 otherwise.if all a,b and c are coprime to 5 RHS would be greater than 5 and a multiple of 5 so we let a=5. Now RHS does not equal 2 and LHS is prime so RHS must be odd so b^4+c^4 is odd. Thus one of b or c is even. Hence we let b = 2. We have 638+c^4=p. Now if gcd(c,3)=1 then c^4mod3=1 and 638+c^4 would be a multiple of 3. Thus c must be 3. We have to now check whether 719 is a prime. Using the fact that if none of the primes less than (719)^0.5 divide the number we conclude that 719 is prime

Let us take cases 1.All of them are even primes i.e two.This is not possible as P will be 45 which is not a prime. 2.Two are even and one is odd,but that is wrong too as you will get an even number. 3.All of them are odd,which is again not possible as sum and difference of 2 odd numbers is even.(A^4+B^4 is even and C^4-3 is also even) 4.The only remaining case is 2 are odd and one is even. Since one number is even it should be 2 so C=2 C^4=16 A^4+B^4+16-3=P A^4+B^4+13=P Now let us look at the last digits of the fourth powers of odd integers. All fourth powers of numbers ending in 1 end with 1, numbers ending in 3 end with 1, numbers ending in 5 end with 5, numbers ending in 7 end with 1, numbers ending in 9 end with 1, Clearly both the numbers can't end in 1 as it would then be a multiple of 5, So either one or both have to be primes ending in 5 which is 5, Again, let's take cases, 1.Both are 5, 5^4+ 5^4+13=1250+13=1263, But 1263 is divisible by 3 so both can not be 5, 2.The only remaining case is that one of the numbers is 5, 5^4+13+A^4=P, 638+A^4=P, We know that any prime number except 3 can be expressed as (3n-1) or (3n+1), Taking mod 3 on both sides, 638mod3=2, Since any prime other than 3 can be expressed as 3n+1 or 3n-1, Taking (3n+1)^4mod 3 , or , (3n-1)^4mod3 ,or (3n-1)^4mod3, (-1)^4mod3,or,1^4mod3, In both the cases it is 1mod 3, Which implies 638mod3+1mod3=0mod 3 But since P is a prime it can't be divisible by 3,and it is obvious that the prime is not 3 So our asuumption that the prime can be expressed as 3n+1 or 3n-1 is wrong So the only remaning case is that the number is 3 So 3^4+5^4+2^4=722 722-3=719