**Puzzle**

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

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

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True

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

ReplyDeleteProof: 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.

This comment has been removed by the author.

ReplyDeletea*a + 26 = (a-2)(a+2) + 30

ReplyDeletewe 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

True.

ReplyDeleten=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)

true

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

ReplyDeleteIf 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.

true

ReplyDeleteIt is true.

ReplyDeletefor that consider a=2(primenumber)

substitute the value of a in a*a+26 then

2*2+26=4+26=30

(WHICH IS NOT A PRIME)

Because 30=1*30

2*15

3*10

5*6

But A num 'a' is said to be a primenumber if it has only two multiples i.e., 1 and a only

If A is even then A^2+26 is even

ReplyDeleteIf A is NOT a multiple of 3 then A^2 is 1 Mod 3 so A^2+26 is a multiple of 3. This leaves only odd multiples of 3 as candidates of which only 3 is prime. 3^2+26=35 which is not prime.