Q: What is -6n 2?

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-5=6n+7 -5-7=6n+7-7 -12=6n -12/6=6n/6 -2=n

So far, the best and most general pattern found is that, over three, all prime numbers are of the form 6n +/- 1. In other words, they're either 6n - 1 or 6n + 1, for some n. Here is why this is true. We could do a proof by contradiction and assume that all the natural numbers greater than or equal to 5 are prime. (of course they are not!) We start with5 which is 6-1. The numbers would then be 6n - 1, 6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4, and 6n + 5 for some natural number n. If it is 6n, then the number is divisible by 6. When it is 6n + 2, the number is the same as 2(3n+1) so it is divisible by 2. Consider 6n + 3, the number is 3(2n+1), so it is divisible by 3. Last look at 6n + 4, the number is divisible by 2, for it's 2(3n + 2). Therefore all numbers of the form 6n, 6n + 2, 6n + 3, and 6n + 4 are not prime. The only possibilities this leaves are 6n - 1 and 6n + 1. This entire thing can be written more elegantly with congruences, but the goal here was simplicity! There are many other patterns in primes. See the attached link to see them.

6n take-away 4 = 2

6n = 12 n = 2

14

factor by grouping 7m(m-2)+6n(2-m)

All non-prime numbers are divisible by prime numbers. Now the smallest to prime numbers are 2 and 3. The next prime number is 5, which is 6*1 - 1. All larger numbers are in the form of one of 6n, 6n+1, 6n+2, 6n+3, 6n+4, 6n+5. Now 6n is divisible by 2 and so cannot be a prime. 6n+2 and 6n+4 are also divisible by 2 and so cannot be prime. 6n+3 is divisible by 3 and so cannot be prime. That leave 6n+1 and 6n+5 as the only two forms than can be prime. Note though that 6n+5 = 6m-1 where m = n+1. So all primes are of the form 2, 3, 6n+1 and 6n-1. And all primes can be divided by primes. The result follows.

6n+3=15 6n+3-3=15-3=12 6n = 12 n=12/6 = 2

If: -6n-2n = 16 Then: n = -2

m^2 6n - 9n2 - 2m = (m - 3n)(m + 3n - 2)

If you mean: -6n-2n = 16 then -8n = 16 and the solution to the equation is n = -2

-6n-2n 16 can be solved mathematically getting n = - 2.