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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.
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