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

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 3 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 96 where only the coefficent, 9m⁴, is tabulated in order to avoid cluttering the table. Columns 3-7 depict the various sequences obtained from the original sequence of column one. The ellipsis is to remind us that the sequences continue ad infinitum with increasing 9 square. 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 = 8, the numbers 9(8)⁴ and 9(4)⁶ are both equivalent after converting both to the non exponential integer 36864. Note that Table I contains only sequences having four as the second number in the coefficient of y along with column 3 containing multiples of nine. Moreover, similar tables containing 5, 6, 7,.. as the second number in exponential form, may also be constructed.

Table of Sequences and Equations

To generate Table I consecutive m terms, with n² being constant at 9, 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 3-7 is 16 and all these sequences are, therefore, geometric progressions. This differs from Part I where both geometric and arithmetic progressions were obtained for the sequences.

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 3 and a second square root taken. For instance, from the last line of Table I column five, given this number without knowing m, its position in column one is calculated as follows:

To determine the next number in the sequence and its position in column one one can use the column equation to calculate its value in exponential form then use method to find its place in column one. For example, the next number after the last entry 9³(4)¹⁰ is 9³(4)¹² and its position in column one is determined as follows:

where, for confirmation, substituting 192 for m affords:

Go to Part II for another general equation and primality testing. Go to Part IV for sequences where n² = 16.
Go back to Part I and homepage.