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mathew3
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An Arithmetic Solution to the Goldbach Conjecture
Prove that any and every even integer >4 may be expressed as the sum of at least some 2 prime integers.
Proof D.
1. We may regard prime integer multiplication as being equivalent to prime integer summation, i.e. 2x3=2+2+2
2. Therefore, given the Fundamental Theorem of Arithmetic, any and every even integer >4 may be expressed as the summation of a series of prime integers, i.e.,
I = Pa+…+Pb+…+Pc+…
where I is any integer >1 and P is some prime integer
3. Any and every even integer must equal the summation of some two odd integers, therefore
E = Oa +Ob
where E is any and every even number and O is some odd integer.
4. Given the FTOA it must also be the case that for any even integer >4
E = Pa+…+Pb+…+Pc+…
5. Therefore for any and every even integer >4
Pa+…+Pb+…+Pc+…= E = Oa +Ob [4]
6. Therefore the sum Oa + Ob must equal a summation of a series of primes.
7. Since there are at least two addends comprising the Oa +Ob summation then each addend is allowed to be a prime number.
8. E, in this case, must meet two conditions:
a. E must be composed of 2 and only 2 odd integers.
b. E must be a summation of primes.
9. In order to satisfy both conditions a and b then it must be the case that the two odd integers, Oa and Ob must sum as primes where
Pa+Pb = E= Oa +Ob [5]
10. Therefore any and every even integer >4 may be expressed as the sum of at least some 2 prime integers.
Prove that any and every even integer >4 may be expressed as the sum of at least some 2 prime integers.
Proof D.
1. We may regard prime integer multiplication as being equivalent to prime integer summation, i.e. 2x3=2+2+2
2. Therefore, given the Fundamental Theorem of Arithmetic, any and every even integer >4 may be expressed as the summation of a series of prime integers, i.e.,
I = Pa+…+Pb+…+Pc+…
where I is any integer >1 and P is some prime integer
3. Any and every even integer must equal the summation of some two odd integers, therefore
E = Oa +Ob
where E is any and every even number and O is some odd integer.
4. Given the FTOA it must also be the case that for any even integer >4
E = Pa+…+Pb+…+Pc+…
5. Therefore for any and every even integer >4
Pa+…+Pb+…+Pc+…= E = Oa +Ob [4]
6. Therefore the sum Oa + Ob must equal a summation of a series of primes.
7. Since there are at least two addends comprising the Oa +Ob summation then each addend is allowed to be a prime number.
8. E, in this case, must meet two conditions:
a. E must be composed of 2 and only 2 odd integers.
b. E must be a summation of primes.
9. In order to satisfy both conditions a and b then it must be the case that the two odd integers, Oa and Ob must sum as primes where
Pa+Pb = E= Oa +Ob [5]
10. Therefore any and every even integer >4 may be expressed as the sum of at least some 2 prime integers.