>Why are we using alternating primes, though?
Was Oddly confused. See >>8748
Definition:Odd Prime (https://proofwiki.org/wiki/Definition:Odd_Prime)
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Apart from 2 itself, all primes are odd.
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So, referring to an odd prime is a convenient way of specifying that a number is a prime number, but not equal to 2.
>If q is the product of the set of small primes and c'=abq = qc, and d'd'+e' = c', and a must appear twice in the first a+1 [e',1,t] elements of [e',1], how do we know what factors will also make up the two values of t where a is a factor of [e',1,t]?
Nice notation! Will work on this.
>I can understand using primes that are only the sum of two squares as an accelerant for certain c, but remove 23 and other primes that end in 11 in binary, the sum of two squares that are odd end in 01
Ah, i c. Also can clearly see '2' as the oddball ending in zero! And won't be able to eliminate in decimal form, as those ending in '3', '7', '9' are all a mixed bag of '01' and '11' in binary form.
2 → 0000000000010
3 → 0000000000011
5 → 0000000000101
7 → 0000000000111
11 → 0000000001011
13 → 0000000001101
17 → 0000000010001
19 → 0000000010011
23 → 0000000010111
29 → 0000000011101
31 → 0000000011111
37 → 0000000100101
…
3881 → 0111100101001
3889 → 0111100110001
3907 → 0111101000011
3911 → 0111101000111
3917 → 0111101001101
3919 → 0111101001111
Will generate new list (5, 13, 17, 29, 37, 41, 53, 61, 73, etc.)
Oh, i c more correlations! (https://oeis.org/A002144)
A002144 Pythagorean primes: primes of form 4n + 1.
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617
COMMENTS
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Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
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Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004
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Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.
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Not only are the squares of these primes the sum of two nonzero squares, but the primes themselves are also. 2 is the only prime equal to the sum of two nonzero squares and whose square is not. 2 is therefore not a Pythagorean prime. - Jean-Christophe Hervé, Nov 10 2013
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The statement that these primes are the sum of two nonzero squares follows from Fermat's theorem on the sum of two squares. - Jerzy R Borysowicz, Jan 02 2019
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The decompositions of the prime and its square into two nonzero squares are unique. - Jean-Christophe Hervé, Nov 11 2013. See the Dickson reference, Vol. II, (B) on p. 227. - Wolfdieter Lang, Jan 13 2015
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p^e for p prime of the form 4*k+1 and e>=1 is the sum of 2 nonzero squares. - Jon Perry, Nov 23 2014
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Primes p such that the area of the isosceles triangle of sides (p, p, q) for some integer q is an integer. - Michel Lagneau, Dec 31 2014
and our primes ending in binary '11'?
A002145 Primes of the form 4n+3. (https://oeis.org/A002145)
3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563, 571
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Or, odd primes p such that -1 is not a square mod p, i.e., the Legendre symbol (-1/p) = -1. [LeVeque I, p. 66]. - N. J. A. Sloane, Jun 28 2008
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Primes which are not the sum of two squares, see the comment in A022544. - Artur Jasinski, Nov 15 2006
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Natural primes which are also Gaussian primes. (It is a common error to refer to this sequence as "the Gaussian primes".)
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Inert rational primes in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
Validating:
3 → 0000000000011
7 → 0000000000111
11 → 0000000001011
19 → 0000000010011
23 → 0000000010111
31 → 0000000011111
43 → 0000000101011
…
499 → 0000111110011
503 → 0000111110111
523 → 0001000001011
547 → 0001000100011
563 → 0001000110011
571 → 0001000111011