User:SHAWWPG19410425/Integer factorizing using binary remainders

The page with the working of two examples on factorization by solving the very straight forward but extremely lengthy equations for the binary remainders [ - 1] or [ + 1] of the prime factors of an uneven composite has been deleted. This is because in the first place, it has proved impossible to find the period or periods of the prime factors so that the equations in the unknown remainders could be written down and in the second place, the integers arising are very large. For example when doing the factorization, Q[6] = 65 [5, 13], the equations contain integers having up to eleven decimal digits. In the case of factorizing say a number like 2021, the number [2021]^{8} occurs and this is of the order 2.56 x 10^{26}.

Factorization by solving for the binary remainders, while having been proved valid has been found to be little more than a mathematical curiosity when applied as above.

There promises to be a good way of factorizing uneven integers that have two prime factors. It has been found that any uneven integer that has two uneven integer factors irrespective of whether these are prime or composite can be expressed as the difference of two perfect squares. When this difference of squares is expressed in terms of unknown binary remainders and an unknown index which is an exponent of two then at this stage it has proved possible so far to solve for these unknowns. The purpose doing examples is to find if there is a pattern enabling the number of binary remainders to solve for to be specified.

The reader is particularly referred to the heading " Deductions " that considers the elementary logic of arithmetic. This section contains worked examples of factorizations. The discussion is in " Factorization/Integers/Binary remainders ".SHAWWPG19410425 (discuss • contribs) 09:03, 10 March 2014 (UTC)

SHAWWPG19410425 (discuss • contribs) 16:21, 18 February 2014 (UTC)SHAWWPG19410425 (discuss • contribs) 15:02, 30 January 2014 (UTC)