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### Math Prime number puzzle 15april

Puzzle

If 'a' is a prime number, then prove, that a*a+26 is NOT a prime number (whether it is true).

1. True! If 'a' is a prime number then a*a+26 will not be a prime number.

Proof: if 'a' is not a prime number then it would not be divisible by 3. After dividing by 3 it gives remainder '1' or '2' then a*a % 3 would be 1 and 1+26=27 that is divisible by 3. so "a*a+26" is not a prime number bcoz it is divisible by 3.

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3. a*a + 26 = (a-2)(a+2) + 30

we can directly check for a equal to 3,

a*a+26 = 35 [NOT PRIME]

for consecutive 5 numbers scenario, consider them as below

a-2, a-1, a, a+1, a+2
O1, E1, O, E2, O2

here O=Odd, E=even

out of O1, E1, and O
at least one is divisible by 3

if O1 is divisible by 3 then (a-2)(a+2) is divisible by 3
if E1 is divisible by 3 then O2 is divisible by 3 and hence (a-2)(a+2) is divisible by 3

so we can write the equation (a-2)(a+2) + 30 as
=> 3K + 30
=> 3(K+10)

thus the number is divisible by 3 if a is not equal to 3, and hence is NOT PRIME

4. True.
n=3:
3*3+26=35=5*7
n=3k+1:
(3k+1)*(3k+1)+26=3*(k(3k+2)+9)
n=3k-1:
(3k-1)*(3k-1)+26=3*(k*(3k-2)+9)

5. Let us divide it by 3. Its a prime number, So dividing it by 3 would result in 1 or 2 as remainder.

If 1 is remainder and 26 is added to it. result would be 27 and it would be divisible by 3 So not a prime number.

If 2 is remainder and 26 is added to it. result is 28 which is clearly not a prime number.

So a * a + 26 is not a prime number.

6. It is true.