mjr

03-30-2015, 05:17 PM

I was pondering this the other night while dozing off, so I wrote a little program to test it out, and got some fascinating results...so much so that I wrote a small document about it. I called it "Summed Sequential Grids and Primality: A Conjecture".

Basically, you take any number. Let's say 2. You square it, and make a grid with that many cells (i.e. 4 cells). Each cell then has a sequential number in it (i.e. you number them 1, 2, 3, etc.).

When you sum them, you get a number. If that number is even, you add 1 to it. If it's odd and NOT prime, you add two to it.

As an example, in our 2 x 2 grid above, the answer when summed is 10. Since 10 is even, adding 1 produces 11, which is prime.

A 3 x 3 grid sums to 45, which is odd, but NOT prime. Adding two gives 47, which is prime.

However, there is another interesting phenomenon here...

Assuming a 5 x 5 grid, the sum is 325, which is odd but NOT prime. So adding two results in a sum of 327. The interesting thing is that 327 is also not prime, but it is divisible evenly by two prime numbers: 3 and 109.

These results hold for larger grids, too. Assume a grid of 3968 x 3968.

Sum: 123,952,898,252,800

Since the value is even, 1 is added, resulting in a value of 123,952,898,252,801, a prime number.

Further, assuming a grid of 3,972 x 3,972 cells, the resulting calculations yield:

Sum: 124,453,464,579,720

This value is even, so per procedure, 1 is added, to produce 124,453,464,579,721. However, 124,453,464,579,721 is a non-prime number, and can be divided by the prime number 10,500,473, which is indeed a prime number. The result of the division is 11,852,177, also a prime number.

The largest result tested was a grid of 10,000 x 10,000. The results were as follows:

Initial Sum: 5,000,000,050,000,000

Even number, so by procedure add 1: 5,000,000,050,000,001

The resultant figure is prime.

Within this methodology, 1,046 of the 10,000 grids tested had a prime sum result.The remaining 8,954 could be divided by a prime number, the quotient of which was also prime.

Basically, you take any number. Let's say 2. You square it, and make a grid with that many cells (i.e. 4 cells). Each cell then has a sequential number in it (i.e. you number them 1, 2, 3, etc.).

When you sum them, you get a number. If that number is even, you add 1 to it. If it's odd and NOT prime, you add two to it.

As an example, in our 2 x 2 grid above, the answer when summed is 10. Since 10 is even, adding 1 produces 11, which is prime.

A 3 x 3 grid sums to 45, which is odd, but NOT prime. Adding two gives 47, which is prime.

However, there is another interesting phenomenon here...

Assuming a 5 x 5 grid, the sum is 325, which is odd but NOT prime. So adding two results in a sum of 327. The interesting thing is that 327 is also not prime, but it is divisible evenly by two prime numbers: 3 and 109.

These results hold for larger grids, too. Assume a grid of 3968 x 3968.

Sum: 123,952,898,252,800

Since the value is even, 1 is added, resulting in a value of 123,952,898,252,801, a prime number.

Further, assuming a grid of 3,972 x 3,972 cells, the resulting calculations yield:

Sum: 124,453,464,579,720

This value is even, so per procedure, 1 is added, to produce 124,453,464,579,721. However, 124,453,464,579,721 is a non-prime number, and can be divided by the prime number 10,500,473, which is indeed a prime number. The result of the division is 11,852,177, also a prime number.

The largest result tested was a grid of 10,000 x 10,000. The results were as follows:

Initial Sum: 5,000,000,050,000,000

Even number, so by procedure add 1: 5,000,000,050,000,001

The resultant figure is prime.

Within this methodology, 1,046 of the 10,000 grids tested had a prime sum result.The remaining 8,954 could be divided by a prime number, the quotient of which was also prime.