While Goldbach’s conjecture is that every even number equal to or greater than 4 can be expressed as the sum of two primes, minus-Goldbach’s conjecture (I call it this way because it’s unnamed but has to be referred to) is that every even number can be expressed as the difference between two primes.
While for every even number the number of corresponding Goldbach’s pairs is finite, the number of minus-Goldbach’s pairs is not (see how easy one can time-jump 107 years ahead and reach the Polignac's conjecture). In order to have the algorithm stop we have to choose an interval, in which a finite number of pairs is calculated (see Step 3 below).
My algorithm for calculating minus-Goldbach’s pairs is following:
Step 1. Choose 2n (any even number that you want expressed as the difference between two primes).
Step 2. Find │2n│ (absolute value of 2n) and its half │n│.
Step 3. To find the primes p1 и p2, the difference of which is 2n, you have to search for them at a distance of │n│ from the natural number x (x=│2n│+i). The values of i are natural numbers between 1 and z. You must also choose the value of z, which is the maximum remoteness of x from │2n│. Thanks to those limitations we are able to calculate a finite number of pairs of primes, otherwise the algorithm wouldn’t be able to stop (see above).
Step 4. For every i=1 to z, calculate x (x=│2n│+i), p2 (p2=x-│n│) and p1 (p1=x+│n│).
Step 5. If p2 and p1 are primes, i.e. if every time when each is divided by 2 or odd numbers lesser than it (excluding 1) there are remainders existing, then output them.
Thanks to Plamen Antonov, you can see the only 'minus-Golbach' calculator on the web here.
While for every even number the number of corresponding Goldbach’s pairs is finite, the number of minus-Goldbach’s pairs is not (see how easy one can time-jump 107 years ahead and reach the Polignac's conjecture). In order to have the algorithm stop we have to choose an interval, in which a finite number of pairs is calculated (see Step 3 below).
My algorithm for calculating minus-Goldbach’s pairs is following:
Step 1. Choose 2n (any even number that you want expressed as the difference between two primes).
Step 2. Find │2n│ (absolute value of 2n) and its half │n│.
Step 3. To find the primes p1 и p2, the difference of which is 2n, you have to search for them at a distance of │n│ from the natural number x (x=│2n│+i). The values of i are natural numbers between 1 and z. You must also choose the value of z, which is the maximum remoteness of x from │2n│. Thanks to those limitations we are able to calculate a finite number of pairs of primes, otherwise the algorithm wouldn’t be able to stop (see above).
Step 4. For every i=1 to z, calculate x (x=│2n│+i), p2 (p2=x-│n│) and p1 (p1=x+│n│).
Step 5. If p2 and p1 are primes, i.e. if every time when each is divided by 2 or odd numbers lesser than it (excluding 1) there are remainders existing, then output them.
Thanks to Plamen Antonov, you can see the only 'minus-Golbach' calculator on the web here.
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