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Polynomial Pronic Alignment

Seen in the Sacks Number Spiral, all polynomials of type N^2 have a fascinating property in common that the traditional parabolic graphing of polynomials (in the Cartesian plane) does not show - namely, pronic alignment on the third coefficient.

Alignment with the pronic axis of the spiral occurs when the number of terms exceeds the third coefficient. So, for example, in the case of n^2+n+41, the alignment occurs on the 42nd term (value 41), when the result of the expression is 1763. This is the first result to have an offset of +41, with the preceding offset being -41, both from the pronic number of 1722. It should also be noted that 1763 is a composite with the prime factors of 41 and 43 - the first two results of the polynomial expression!

The pronic offset number of the polynomial is simply determined by the sign and number of the third coefficient of the polynomial. A negative coefficient produces a negative offset. To better understand how this works, try any N^2 polynomial in Vortex. Or to see how it works in QTest, press the O key when you click Generate. Before this offset stabilizes, the polynomial moves from positive to negative offset and back again. This appears chaotic, but when viewed in the perfect square-aligned (and hence, pronic-aligned) spiral, this is in fact an orderly rotation that eventually flattens out, paralleling the pronic axis. Larger constants mean larger offsets that mean more rotations.

It appears that without exception the greatest concentration of primes for any polynomial - including the uninterrupted prime sequence of a perfect prime polynomial like n^2+n+41 - occurs before it aligns with the pronic axis.

Terms Result Offset Pronic  
0
41-P
-1
42

Graph of the offset from the nearest pronic number showing oscillating values
before stabilizing on offset 41.

(Note that the pronic axis is represented by the zero axis in this graph. To gain a proper understanding of how numbers are distributed along offsets, see and try the Sacks Number Spiral .)

1
43-P
1
42
2
47-P
5
42
3
53-P
-3
56
4
61-P
5
56
5
71-P
-1
72
6
83-P
-7
90
7
97-P
7
90
8
113-P
3
110
9
131-P
-1
132
10
151-P
-5
156
11
173-P
-9
182
12
197-P
-13
210
13
223-P
13
210
14
251-P
11
240
15
281-P
9
272
16
313-P
7
306
17
347-P
5
342
18
383-P
3
380
19
421-P
1
420
20
461-P
-1
462
 
21
503-P
-3
506
 
22
547-P
-5
552
 
23
593-P
-7
600
 
24
641-P
-9
650
 
25
691-P
-11
702
 
26
743-P
-13
756
 
27
797-P
-15
812
 
28
853-P
-17
870
 
29
911-P
-19
930
 
30
971-P
-21
992
 
31
1033-P
-23
1056
 
32
1097-P
-25
1122
 
33
1163-P
-27
1190
 
34
1231-P
-29
1260
 
35
1301-P
-31
1332
 
36
1373-P
-33
1406
 
37
1447-P
-35
1482
 
38
1523-P
-37
1560
 
39
1601-P
-39
1640
 
40
1681
-41
1722
 
41
1763
41
1722
 
42
1847-P
41
1806
 
43
1933-P
41
1892
 
44
2021
41
1980
 
45
2111-P
41
2070
 
46
2203-P
41
2162
 
47
2297-P
41
2256
 
48
2393-P
41
2352
 
49
2491
41
2450
 
50
2591-P
41
2550
 
51
2693-P
41
2652
 
52
2797-P
41
2756
 
53
2903-P
41
2862
 
54
3011-P
41
2970
 
55
3121-P
41
3080
 
56
3233
41
3192
 
57
3347-P
41
3306
 
58
3463-P
41
3422
 
59
3581-P
41
3540
 
60
3701-P
41
3660
 
 



   
         

 

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