Prime numbers from Pythagorean triples
We know every odd number can be a number of Pythagorean triple. Few Pythagorean triples for odd numbers from 3 in the natural series are
3 4 5
5 12 13
7 24 25
9 40 41
11 60 61
13 84 85
15 112 113
All the edge numbers (smallest and biggest) are odd while the central numbers are all even. The general form of this series of Pythagorean triple is [ (2n+1), 4Σn , 1 + 4 Σ n ]
Now take a central number from any one of the Pythagorean triple (say y’) in (x,y,z).Starting with y, generate an arithmetic progression with the smallest number x as the common difference. It will be in the form of (y + nx) where n = 1,2,3,4,..... Few such series are given in Table.1.
Table.1. Pythagorean triples and its arithmetic progressions
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Pythagorean triple general form series
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(3,4,5) (4 + 3n) 4,7,10,13,16,19,22........
(5,12,13) (12 + 5n) 12,17,22,27,32,37,42,....
(7,24,25) (24 + 7n) 24,31,38,45,52,59,66.....
(9,40,41) (40 +9n) 40,49,58,67,76,85,94,....
(11,60,61) (60+11n) 60,71,82,93,104,...........
(13,84,85) (84 +13n) 84,97,110,123,.............
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Since there are infinite odd numbers, there will be infinite Pythagorean triples and hence infinite such series. It is noted that some numbers may be existing in two or more series (for example, in the first two series 22 is found to be common),while some other numbers may not be found at all in any one of the series. If N is such a missing number, then it generates the prime by (2N +1). For example, below 50, the missing numbers are 1,2,3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36,39,41,44,48........ They produce all the prime numbers from 3 up to 97.
With the general form of Pythagorean triple, we can express the series as
4Σ m + (2n +l) n
Where m =1,2,3,4..... and n = 0,1,2,3,...... Hence all the numbers in all the series can be represented simply by 2m (m+n+1) + n. Thus the numbers that cannot be derived by varying m and n values give prime numbers on addition of 1 with twice of them.
Let N be a number in the series, then N = 2 m (m+n+1) + n ,where m = l,2,3,4... and n = 0,1,2,3,... The values of N are quantized. The forbidden values can be determined that fall in the gap between the two successive allowed values. Table.2 exemplifies this feature.
Table.2. Missing values of the series and prime numbers
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m n N(allowed) N(forbidden) prime
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1 0 4 1,2,3 3,5,7
1 1 7 5,6 11,13
1 2 10 8,9 17,19
2 0 12 11 23
1 3 13 ..... .....
1 4 16 14,15 29,31
2 1 17 ...... .....
1 5 19 18 37
2 2 22 20,21 41,43
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