4.0 Conclusions.

The two methods described in this paper have produced quadratics that generate all the 240 prime numbers between 0 and 1500. By fully exploring the flexibility of the extended Ulam's Spiral, and performing extended searches with the Composite Search method, it is believed that many more could be found. Subsequently this would enable the possibility of finding a pattern, if one exists, which would permit further quadratics to be predicted.

The highest prime number generated to date by a quadratic found in this exercise is 15,319. To extend significantly beyond this would require a much faster computing capability and a much larger Ulam Spiral, probably one that would have to be scrolled. Also, coding in a high level language, to produce a stand alone EXE file would be essential for very large numbers to be handled.

Finally, in the next paper on this subject, the first pattern in these quadratics will be described. This paternation will allow their prediction, rather than the necessity for searching, and will establish sets of "families" of quadratics, which will permit the generation of a possibly limitless number of prime number sequences extending into very high values.

M6 Version 1.0.0
Ó P.G.Bass, December 2011

APPENDIX A.

An Analytical Evaluation of Ulam's Spiral.

An analytical representation is shown in Fig. A.1 below.

Fig. A.1 - Ulam's Spiral, Analytical Representation.

In Fig. A.1 the following parameters are defined as follows:-

(i) P is the Base Number in the Base Cell.

(ii) P+nk are the values in all other Cells.

(iii) k is the Step Size.

(iv) The cells are numbered 1, the Base Cell, to 25 etc around the spiral.

(v) R_n is the Spiral Ring number, R₁ contains just the Base Cell, (R₁ = 0), R₂ contains cells 2 to 9, (R₂ = 1), R₃ contains cells 10 to 25, (R₃ = 2) etc.

(vi) D is the direction value in which successive terms of a quadratic lie. D = - 3 to + 4 for directions A to H, (as in Table 2.1).

(vii)B is the number of spiral boundaries crossed by successive quadratic terms.

(viii) n_r is the number of cells counting from the first cell in a ring, to the cell containing the value of interest. Note the first cell in R₁ is #1, in R₂ it is #9 and in R₃ it is #25 etc, as shown in Fig. A.1.

Analysis for a₂.

a₂ is only affected by k and B. The values of a₂, determined from a series of term sequences in the spiral, for a number of values of k and B, are shown in the following table.

k	B	a₂

1	1	4
1	2	16
1	3	36
2	1	8
2	2	32
2	3	72
3	1	12
3	2	48
3	3	108

Table A.1 - Values of a₂ for k and B.

From Table A.1 it is clear that

a₂ = 4kB²

(A.1)

Analysis for a₁.

a₁ is affected by k, B, D, and R_n. The values of a₁ for various values of these parameters, determined from the spiral values of quadratic sequences, are shown in Table A.2 below. R_n here is the ring number containing the spiral cell in which the first quadratic term resides.

k	B	R_n	D	a₁	k	B	R_n	D	a₁	k	B	R_n	D	a₁

1	1	0	A	-3	1	1	1	A	4	2	1	1	A	10
			B	-2				B	6				B	12
			C	-1				C	7				C	14
			D	0				D	8				D	16
			E	1				E	9				E	18
			F	2				F	10				F	20
			G	3				G	12				G	22
			H	4				H	14				H	24
1	2	0	A	-6	1	2	1	A	10	2	2	2	A	52
			B	-4				B	12				B	56
			C	-2				C	14				C	60
			D	0				D	16				D	64
			E	2				E	18				E	68
			F	4				F	20				F	72
			G	6				G	22				G	76
			H	8				H	24				H	80
2	3	2	F	108	3	2	2	D	96	3	2	3	A	126
			G	114				E	102				B	132
			H	120				F	108				C	138

Table A.2 - Values of a₂ for k, B, R_n and D.

Note that some of these values may vary if the term sequence crosses the Base Number boundary.

Analysis of Table A.2 shows that a₁ can be represented by

a₁ = ( 8R_n + D )kB

(A.2)

Analysis for a₀.

a₀ is dependent upon k, P, R_n and n_r. Table A.3 below shows the values of a₀ for various values of these parameters as determined from the spiral.

k	R_n	n_r	P	a₀

1	1	5	1	5
1	1	7		7
1	1	9		9
2	1	3		5
2	2	9		33
2	2	14		43
3	2	4		34
3	2	10		52
3	2	15		67

Table A.3 - Values of a₀ for k, R_n, n_r and P.

Analysis of Table A.3 shows that

a₀ = [ 8å

^{n - 1}_{n = 1}

( R_n ) + n_r - 1 ]k + P

(A.3)

which via the standard formula for arithmetic series, becomes

a₀ = ( 4( R_n-1 )R_n + n_r -1 )k+P

(A.4)

Note that (A.4) can be used to determine the value of the contents of any cell in the spiral.

The final equation governing the generation of quadratics from Ulam's Spiral is, by combining (A.1), (A.2) and (A.4)

N = 4kB²x² +( 8R_n + D )kBx +( 4( R_n - 1 )R_n +n_r - 1 )k + P

(A.5)

Thus if k = 0.25, B = 1, R_n = 0, (so that n_r = 1), D = 4, (Direction H) and P = 41, (A.4) becomes

N = x² + x + 41

(A.6)

i.e. Euler's equation.

Similarly if k = 0.5, B = 1, R_n = 0, (so that n_r = 1), D = 0, (Direction D) and P = 29, (A.5) becomes

N = 2x² + 29

(A.7)

Legendre's equation.

M6 Version 1.0.0
Ó P.G.Bass, December 2011

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