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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