87 replies to this topic

[Wizzbang]

I just saw this puzzle today and see it a bit differently.

We all know that 2 is the only even prime number.
We also know that an odd number times an odd number is always an even number. The same is true for multiplying even numbers (even x even = even)

So...

(A*A)+26 can be seen as (ODD*ODD)+EVEN or (EVEN*EVEN)+EVEN. Either way the result is always even and always greater than 2. As such, this expression can never yield a prime result. The answer will always be non-prime.

Edited by Wizzbang, 09 July 2008 - 12:42 PM.

[cmabb21]

We also know that an odd number times an odd number is always an even number.

I have to disagree. An odd number, say 5 times 5.. is not an even number...
5*5=25
also
7*7=49

last time i checked, 25 and 49 were not even numbers...
im pretty sure its all odd numbers times an odd number equals an odd number and all even numbers times an even number equals an even number.

Edited by cmabb21, 09 July 2008 - 03:24 PM.

[toumba]

if a was any number, and you squared it, it is guaranteed to have divisible factors, making it composite(i.e 11*11=121 with factors 1, 11, 121. 7*7=49 with factors 1, 7, 49. 5*5=25 with factors 1, 5, 25). So even if a*a is an odd number there will be at least 3 factors to it making it a composite number and not prime. if a happened to be 1, then 1*1+26 would be 27 which is a composite number with factors 1,3,9, and 27.

Edited by toumba, 16 July 2008 - 09:46 PM.

[Prime]

Sorry, I stumbled upon this one so late.
Here is a SIMPLE PROOF:

Any prime number P may be expressed as P=3*n + 1 or P=3*n+2. (Another way of saying prime gives a remainder of 2 or 1, when divided by 3).
Thus P² + 26 can be expressed as 9n²+6n+27 or 9n²+12n+30. Both expressions are wholly divisible by 3.
Therefore, P² + 26 is always divisible by 3.

QED

[Prime]

Any prime number P may be expressed as P=3*n + 1 or P=3*n+2. (Another way of saying prime gives a remainder of 2 or 1, when divided by 3).

Technical correction: Any prime number, but 3 -- that is. For 3 we have to check separately: 3² + 26 = 35 -- not a prime. Everyting else is divisible by 3.

[iangardner777]

Hey all, I just skimmed through this topic and saw some pretty complicated math. While quite a few people got to the correct answer I have a way that I think is a good bit simpler than anything I saw. So, just in case anybody is interested…

(A*A) + 26 is never prime if A is prime.

The key to my way is modular arithmetic. Sense, A is prime we know that it is not divisible by 3. Therefore, A modulo 3 = 1 or 2. (This means that A / 3 has a remainder of 1 or 2.) Let’s split this into two cases.

*It is important to know that modular arithmetic works the same as regular arithmetic when dealing with addition and multiplication.

Case 1: A modulo 3 = 1: Thus A*A modulo 3 = 1*1 = 1

Case 2: A modulo 3 = 2: Thus A*A modulo 3 = 2*2 = 4 = 1 modulo 3**

**Here I used the fact that 4 and 1 are equal with respect to modulo 3. That is because both of them are 1 more than a multiple of 3. This is the definition of modulo arithmetic.

From the two above cases we have proved that any A*A = 1 modulo 3 when A is prime. Now by adding any number that equals 2 modulo 3 (which is equal to -1 modulo 3; they are both 1 less than a multiple of 3) we see that we will get a number that equals 0 modulo 3. In other words the number will be divisible by 3 and thus not prime! I saw bonanova touched on this, 26 is completely arbitrary. Any number that equals -1 modulo 3 will work.

Sorry, I didn’t explain this very well, but it really is quite simple when you think of it in these terms. (And really sorry if someone already gave this process I just skimmed the first few pages. Hmmm... and this is basically what prime just said, but oh well I typed it all out now. He is right though 3 will have to be tested seperately, but since 35 is not a prime we're good.)

[qwertyuiop]

Spoiler for solution

a=3
3*3=9
9+26=35
35 is not prime

[nobody]

Spoiler for easy proof

Showing that a*a+26 is not prime when a=3 is true but not a math. proof. However it is quite easy: (if I didn't misunderstand)
a*a + 26 =
a*a -1 + 27=
(a-1)*(a+1) + 27

Now, since a is prime, it can not be a mutliplier of 3. Then either (a+1) or (a-1) should be a multiplier of 3.
Then (a-1)*(a+1) is multiplier of 3
And (a-1)*(a+1) +27 is also multiplier of 3
Then it's not prime.

[ikillyou]

It might be harder than that. If 26 were a multiple, but it's only added.
I'm at a loss on this one.

I'm not sure if your equation is right but theres an easier way. the smallest non prime number that is undebateable (1 is debateable) is 4.

4*4 is the same as 4x4x4x4 which is 256. add 26 to that and you have 282 which can be divided by 2 and thus not prime.

[zeepuzzlemaster]

I'm not sure if your equation is right but theres an easier way. the smallest non prime number that is undebateable (1 is debateable) is 4.

4*4 is the same as 4x4x4x4 which is 256. add 26 to that and you have 282 which can be divided by 2 and thus not prime.

Well, 4^4 = 4 x 4 x 4 x 4

But 4*4 = 4 x 4