A New General Equation and Sequences for Sophie Germain Type Identities (Part IV)

Table of Sequences Generated from the New Equation

Sophie Germaine during the 1800's came up with the identity relation which bears her name and is featured in a Wikipedia article. The identity is a polynomial factorization which states that:

I have found that this identity is one of an infinite number of equations which are generated from this particular general equation:

where m = 1 and n = 2 corresponds to the original Sophie Germain identity. This general equation is being used here to generate mainly a series of sequences, where the sequences for m = 1 and n = 2 were previously generated in Part I. Another equation similar to this one which applies to Sophie Germain identities and may be used for primality testing is covered in Part II.

To generate the sequence on this page n is set equal to 4 and the m values are incremented to values greater than zero. This generates an infinite number of identity equations as shown in column one of Table I for m up to 80 (and a single value of 192) where only the coefficent, 16m⁴, is tabulated in order to avoid cluttering the table. Columns 3-6 and 8-10 depict the various sequences obtained from the original sequence of column one. The ellipsis is to remind us that the sequences continue ad infinitum where the values in the equations either increase by odd m² (columns 4-6) or increase by 16² (columns 3, 8-10). In addition, a value of k ≥ 0 is used as the exponent variable for each of the sequences.

To generate the terms in the table we use the equations posted in the first row along with the k value starting at zero. The value obtained corresponds to one of the entries in column two. While sometimes these two numbers might differ, as for example when m = 3, the numbers 16(3)⁴, 16(9)² and 16³ are equivalent after converting them to the non exponential integer 4096. Note that the equations of columns 4-6 of Table I differ from that of Part I in that the exponents of 16 are k+1 instead of 2k+1 and, therefore, give consecutive multiple exponents of 16.

Table of Sequences and Equations

To generate Table I consecutive m terms, with n² being constant at 16, are used in the general equation to produce the coefficients of y⁴.

Calculation of Differences Between Adjacent Terms in a Sequence

The difference between terms in columns 4-6 is 16 and all these sequences are, therefore, geometric progressions, while columns 3 and 8-10 are arithmetic progressions, similar to Part I.

Generation of m from a Sequence Entry

Finally, given a term in the sequence or one further down into the sequence whose m is not known, one can determine where in column one to place m. The numbers in exponential form are first converted into an integer, its square root taken, followed by division by n = 4 and a second square root taken. For instance, from m = 80, column 9 of Table I given the next number in the sequence one can calculate m, its position in column one, as follows:

Alternatively, one can take the 4th root of 2517630976 and then divide by n = 2 and arrive at the same answer:

where, for confirmation, substituting 112 for m affords:

Go to Part II for another general equation and primality testing.
Go back to Part III and homepage.