Goldbach's Theorem: Its Proof
Dr. S.K.Kapoor has authored a book
"Goldbach Theorem" in which proof of Goldbach's conjecture has been published as
"Proof of Goldbach Theorem". Subsequently Step 2A has been added. This proof
inclusive of Step 2A is published as an Article in
Issue
No. 10 (July 2000) of Vedic Mathematics Newsletter
Precisely the conjecture is that
every even greater than 2 can be written as a pair of sum of primes. It is part
of the famous letter (June 7, 1942) of Christian Goldbach (1690-1764) to great
Swiss mathematician Leonhard Euler. Since then this conjecture has remained a
brain teaser and unsolved problem of the order of Fermat's Last Theorem and
Riemann Hypothesis.
The book "Goldbach Theorem" has
four chapters. The proof except Step 2A has been settled in Chapter 1 of the
book. All what is presumed as background is only the mature comprehension of the
whole numbers. As every even number by definition admits two parts, therefore,
it is this property of evens which has come very handy to express even E as M+M.
With this the maximum possible pairs with numbers 0 to E have been formed
putting a restriction that some of the numbers of the pairs should always be
equal to E. The pairs are designated as duplexes. Then the recursive subsets of
the source set of maximum number of duplexes of given even number E are
constructed and the subset of duplexes with both numbers of the duplexes being
primes is reached at. The cardinality of this set is computed as
³
1/4×ÖE.
As 1/4×ÖE for E
³
64 is
³ 2, so after accounting for the duplex (1,E-1), there
always remains a minimum of one such duplex whose both members are to be primes
and with it the conjecture stands satisfied. Otherwise, this in fact, amounts to
extension of the conjecture for E³
64 from expectation of one
partition for E as sum of primes to that of the minimum of 1/4×ÖE number of partitions for E as sum of primes.
Chapter 2 of the book takes us to
Vedic geometric inspiration and approach of di-monad format (a format which
accepts entity as of two parts). The interesting property of the geometric setup
of "square" of area E is that such a square would be admitting
ÖE as a side of the square. This gives us the ratio of area
E and the sum of its four sides as 1/4×ÖE. The conceptual parallelism is much inspiring. This makes
Vedic geometric approach as much potentialised approach.
Chapter 3 of the book introduces a
slide rule for finding out solutions for E=P+Q. This together with the computer
testing (Chapter 4) for the number of solutions of equation E=P+Q adds empirical
value to the proof.
Link for proof:
PROOF
OF GOLDBACH THEOREM