The Goldbach pairs (Goldbach partitions), when considered multiplicatively rather than additively, can be used to generate what are known as odd semiprimes - the products of two odd prime factors (unique or identical). When you realize that fact, then you may make the conjecture that every perfect square integer can be expressed as the sum of an odd semiprime and another perfect square. We know that odd times odd will always generate odd, and we also know that the sequence of odds sum progressively to the sequence of perfect squares. We also know that the integers 0 and 1, the so-called identity elements, are also perfect squares. We also know that another way of stating the strong Goldbach conjecture is to say that every integer n is the average of two odd primes - hence, the Goldbach relates to midpoint. So we have the introduction of perfect squares and midpoint into the Goldbach conjecture which leads to a conclusion that the Goldbach conjecture has much deeper roots than the simple statement first suggested by Christian Goldbach. The conjecture relates to all the group axioms of arithmetic, along with the principles of elementary calculus and the principles of Euclidian geometry and the bilateral symmetry of the Cartesian grid.
I would suggest that the true gist of Goldbach lies in the principles of universality and reversibility that underlie the entire fabric of human logic and mathematics.
Example: Perfect square 25 = perfect square 4 plus semiprime 21=prime 3 x prime 7. Try a few billion others ( and don't forget those identity elements are also perfect squares)
And this all can be seen by using the group axioms and operations to see that every even integer can be rearranged as a model of the standard numberline. for example, the even integer 8:
0 1 2 3 4 5 6 7 8
8 7 6 5 4 3 2 1 0
Examine sums, products, differences of the binary columns along with the differences in the respective rows of those sums, products, and differences. The midpoint generates a perfect square and the Goldbach partitions generate semiprimes. And their differences are always (necessarily) another perfect square. When the Goldbach pairs are examined as the factors of semiprimes rather than as the addends of even integer sums and that those odd semiprimes are part of the sequential production of perfect squares - the Goldbach conjecture is far more tractable.
Play with it.
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