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

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

By varying n for values greater than zero an infinite number of identity equations are generated as shown in column two of Table I where only the coefficent, 4n⁴, is tabulated in order to avoid cluttering the table. If we scroll down column 2 the numbers in each row of column 2 appear to form a complicated sequence of entries based on four to some power but is actually a sequence composed of the entries 4(1), 4(16), 4(81), 4(256), ..., viz., the sequence based on 4(n⁴).

We may, however, visualize the entries of column one table I in a different manner. They may be separated into odd and even n, with the odd n going into column 3 and the rest into columns 4 thru 12. The even n are subsequently separated into what appears to be an infinite number of sequences, one of a type 4^(2k+1)((2j+1)²)² an a second of a type 4^2j+1((2k+1)²)² where 2j+1 is constant for each sequence, and with j starting at zero for the first sequence and incrementing by one after that for each successive sequence. Moreover, although the former equation generates entries that are associated with even n, the first entry starts off as odd n. Table I depicts the first eight sequences where an infinite number of sequences of the type 4^(2k+1)(2j+1²)² and 4^2j+1((2k+1)²)² follow after the ellipses.

In addition, only the coefficients of y⁴ from x⁴ + 4n⁴y⁴ are shown in the table again to avoid cluttering. For all equations derived from an even n, a value of k ≥ 0 is used as one of the variables within a particular sequence. It is to be noted that 4((2k+1)²)² is equal to 4(n²)² when j = 0 (column 3, Table I) and, therefore, this sequence although considered even n can be derived from odd n.

Not all values are listed in the table, just enough to get a flavor for the way the sequences take on certain values. To generate the terms in the table, we can either take the value of n⁴ and divide repeadedly by four to get four to some power (where 1 is considered a power) and an odd number to a second power, group all the fours to some powers together, then combine the group of four and the odd number into one expression. Or we can take the entire coefficient of y⁴ obtained from the general equation and carry out the same repeated division. Either method gives two numbers in exponential form that can be used to determine which sequence, viz., table column, to use.

For instance, for an odd n:

and for an even n:

Table of Sequences and Equations

To generate the table, consecutive n terms, are used in the general equation to produce the coefficients of y⁴. These coefficients are converted by the methods listed above into the expressions listed in the second column of Table I. Its hard to see a pattern, but upon close inspection there appear to be many similar-like terms which appear to be linked together into interleaved sequences. Consequently, after selecting and grouping these terms together into their appropriate columns, the next step is to find the equations that will generate these terms. The table header of Table I shows the equations that were developed to fit the entries of each particular column. Moreover, in columns 4-6 the part containing the odd numbers to some power is held constant, while in columns 8-11 as well as column 3, it is the part containing the fours to some power that is held constant.

Calculation of Differences Between Adjacent Terms in a Sequence

To calculate the difference between terms in columns 4-6 would be to take adjacent terms and subtract term1 from term2. However, because of the large numbers produced due to the large exponents, a different approach was taken. Every term(2) is divided by the previous term(1) giving a value of 16 for each quotient and these particular sequences are, therefore, geometric progressions. The same method, however, cannot be performed on columns 8-11 since non integer values for the quotients are the result.

Instead the difference (Δ) between terms in a sequence is used (seven terms used in this example), producing a final ΔΔΔΔ in which every quotient is 384 as shown in Table II and, therefore, these sequences are arithmetic progressions. (Method adopted from Table V of Pythagorean multiples). To simplify the calculations the 4^2j+1 is factored out and only the odd numbers squared are used in column one. After the final calculations the 4^2j+1 can be remultiplied with each of the Δ values to give its actual value.

Generation of n from a Sequence Entry

Finally, given a term in the sequence or one further down into the sequence whose n is not known, one can determine where in column one to place n. The exponential number 4^a(odd Number)^b, where a and b are integer exponents, is first converted into an integer, its square root taken, followed by division by 2 and a second square root taken. For instance, from column 4 of Table I the next number in the sequence is 4¹⁵ :

The same number can be obtained directly from inspecting the sequence in the table. Sometimes its easier to carry out the computation than to check to see if n is in the table. For instance, the next number after 2401 with n = 98, expression 4³(6561)² equal to 2754990144 in which n is calculated to be 162. Note that these two numbers 2401 and 6561 may further be reduced to 49² and 81², respectively.

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