Friday, October 26, 2007

Prime number game

I was asked to prove (n2 - 1) will always be divisible by 24 where n being a prime number greater or equal to 5. Here you go: A prime number will always be an (even number + 1) or an (even number - 1). The even number can be replaced with 2*x and the number becomes: (2*x + 1) or (2*x - 1). For n greater than 5, x > 3 and the formula will now be (6*x + 1) or, (6*x - 1) [Not the same x though]. Lets go back to the original question and replace (n2 - 1):

#For (6x + 1) -
((6x + 1)2 - 1) = (36x2 + 12x + 1 -1) = 12x(3x + 1). For any number 'x' either x or (3x+1) will be even. This means no matter what x(3x + 1) is, it will be equal to 2*M where M could be any number. So, (n2 - 1) now becomes 24*M which is always divisible by 24.

# For (6x - 1) -
((6x-1)2 -1) = (36x2 - 12x + 1 - 1) = 12x(3x-1). Same thing again, for any 'x' either x or (3x - 1) will be an even number and 12x(3x - 1) = 24*L where L could be any whole number. Either way, the number is still divisible by 24.

5 comments:

Pri said...

heh. i've never seen this on a blog before. but thats probably cause you and i read a completely different set of blogs.

im reminded of my 7th std tuition teacher mrs.sheshadri. she made me fall in love with mathematics, which no one especially my parents believed could happen.

ashish said...

Whats the mystery behind the (6x+1)/((6x-1)? Any integer can be a form of (6x + y) for y in {0-5}.
for y = 0 its 6x -(not prime)
y = 1 -> (6x + 1) - may be
y = 2 -> 2(3x + 1) - Not
y = 3 -> 3(2x + 1) - Not
y = 4 -> 2(3x + 1) - Not
y = 5 -> (6x + 5) - May be

for any x it will be some number (m - 1) isn't it? So, the form of (6x + 5) will be (6m - 1). To combine our analysis then, the prime numbers can only be of the form of either (6x+1) or (6x-1).

Krishna said...

i know prime numbers are infinite series but can you obtain a function for this series?

ashish said...

I wish there was. So far no equation exist. I know one story when a person wrote a prime number 32 lines of length on a black board, everyone clapped and no one asked how he came up with that number. Seems it was just a fortunate incidence.

ashish said...
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