Search: a070151 -id:a070151
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A002144
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Pythagorean primes: primes of the form 4*k + 1.
(Formerly M3823 N1566)
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+10
482
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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
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OFFSET
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1,1
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COMMENTS
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Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
These are the prime terms of A009003.
-1 is a quadratic residue mod a prime p if and only if p is in this sequence.
If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2 = p (mod q), x^2 = q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004
Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.
The square of a(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = a(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005
Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk, Aug 10 2006
The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi, Mar 28 2008
If we take 4 numbers: 1, A002314(n), A152676(n), A152680(n) then multiplication table modulo a(n) is isomorphic to the Latin square:
1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
and isomorphic to the multiplication table of {1, i, -i, -1} where i is sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i. 1, A002314(n), A152676(n), A152680(n) are subfield of Galois field [a(n)]. (End)
Primes p such that the arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes: this one and A002145. - Ctibor O. Zizka, Oct 20 2009
Equivalently, the primes p for which the smallest extension of F_p containing the square roots of unity (necessarily F_p) contains the 4th roots of unity. In this respect, the n = 2 case of a family of sequences: see n=3 (A129805) and n=5 (A172469). - Katherine E. Stange, Feb 03 2010
k^k - 1 is divisible by 4*k + 1 if 4*k + 1 is a prime (see Dickson reference). - Gary Detlefs, May 22 2013
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
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
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
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
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
This is the set of all primes that are the average of two squares. - Richard R. Forberg, Mar 01 2015
Numbers k such that ((k-3)!!)^2 == -1 (mod k). - Thomas Ordowski, Jul 28 2016
In addition to the comment from Jean-Christophe Hervé, Nov 10 2013: All powers as well as the products of any of these primes are the sum of two nonzero squares. They are terms of A001481, which is closed under multiplication. - Klaus Purath, Nov 19 2023
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REFERENCES
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David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
L. E. Dickson, "History of the Theory of Numbers", Chelsea Publishing Company, 1919, Vol I, page 386
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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p^2 - 1 = 12*Sum_{i = 0..floor(p/4)} floor(sqrt(i*p)) where p = a(n) = 4*n + 1. [Shirali]
Product_{k>=1} (1 - 1/a(k)^2) = A088539.
Product_{k>=1} (1 + 1/a(k)^2) = A243380.
Product_{k>=1} (1 - 1/a(k)^3) = A334425.
Product_{k>=1} (1 + 1/a(k)^3) = A334424.
Product_{k>=1} (1 - 1/a(k)^4) = A334446.
Product_{k>=1} (1 + 1/a(k)^4) = A334445.
Product_{k>=1} (1 - 1/a(k)^5) = A334450.
Product_{k>=1} (1 + 1/a(k)^5) = A334449. (End)
Product_{k>=1} (1 + 1/A002145(k)) / (1 + 1/a(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/A002145(k)) / (1 - 1/a(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log((2*n*s)! * zeta(n*s) * abs(EulerE(n*s - 1)) / (Pi^(n*s) * 2^(2*n*s) * BernoulliB(2*n*s) * (2^(n*s) + 1) * (n*s - 1)!))/n, s >= 3 odd number. - Dimitris Valianatos, May 21 2020
Legendre symbol (-1, a(n)) = +1, for n >= 1. - Wolfdieter Lang, Mar 03 2021
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EXAMPLE
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The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
p a b t_1 c d t_2 t_3 t_4
---------------------------------
5 1 2 1 3 4 4 3 6
13 2 3 3 5 12 12 5 30
17 1 4 2 8 15 8 15 60
29 2 5 5 20 21 20 21 210
37 1 6 3 12 35 12 35 210
41 4 5 10 9 40 40 9 180
53 2 7 7 28 45 28 45 630
...
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MAPLE
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a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a), 4*n+1]; fi; od: A002144 := n->a[n];
# alternative
option remember ;
local a;
if n = 1 then
5;
else
for a from procname(n-1)+4 by 4 do
if isprime(a) then
return a;
end if;
end do:
end if;
end proc:
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MATHEMATICA
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Select[Prime[Range[150]], Mod[#, 4]==1&] (* Harvey P. Dale, Jan 28 2021 *)
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PROG
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(Haskell)
a002144 n = a002144_list !! (n-1)
a002144_list = filter ((== 1) . a010051) [1, 5..]
(Magma) [a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014
(PARI) select(p->p%4==1, primes(1000))
(PARI)
A002144_next(p=A2144[#A2144])={until(isprime(p+=4), ); p} /* NB: p must be of the form 4k+1. Beyond primelimit, this is *much* faster than forprime(p=..., , p%4==1 && return(p)). */
A2144=List(5); A002144(n)={while(#A2144<n, listput(A2144, A002144_next())); A2144[n]}
(Python)
from sympy import prime
A002144 = [n for n in (prime(x) for x in range(1, 10**3)) if not (n-1) % 4]
(Python)
from sympy import isprime
(SageMath)
def A002144_list(n): # returns all Pythagorean primes <= n
return [x for x in prime_range(5, n+1) if x % 4 == 1]
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CROSSREFS
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Cf. A002145, A002314, A002476, A002972, A002973, A003658, A004431, A007519, A010051, A016813, A076339, A094407.
Cf. A004613 (multiplicative closure).
Apart from initial term, same as A002313.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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A002330
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Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0462 N0169)
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+10
39
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1, 2, 3, 4, 5, 6, 5, 7, 6, 8, 8, 9, 10, 10, 8, 11, 10, 11, 13, 10, 12, 14, 15, 13, 15, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20, 25, 22, 26, 28, 25
(list;
graph;
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OFFSET
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1,2
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REFERENCES
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A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
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FORMULA
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EXAMPLE
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The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
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MAPLE
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a := []; for x from 0 to 50 do for y from x to 50 do p := x^2+y^2; if isprime(p) then a := [op(a), [p, x, y]]; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the triples in "trans"
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MATHEMATICA
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Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* Jean-François Alcover, Jul 05 2011 *)
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PROG
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(PARI) f(p)=my(s=lift(sqrt(Mod(-1, p))), x=p, t); if(s>p/2, s=p-s); while(s^2>p, t=s; s=x%s; x=t); s
(PARI) do(p)=qfbsolve(Qfb(1, 0, 1), p)[1]
(PARI) print1(1); forprimestep(p=5, 1e3, 4, print1(", "qfbcornacchia(1, p)[1])) \\ Charles R Greathouse IV, Sep 15 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A002331
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Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0096 N0033)
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+10
23
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1, 1, 2, 1, 2, 1, 4, 2, 5, 3, 5, 4, 1, 3, 7, 4, 7, 6, 2, 9, 7, 1, 2, 8, 4, 1, 10, 9, 5, 2, 12, 11, 9, 5, 8, 7, 10, 6, 1, 3, 14, 12, 7, 4, 10, 5, 11, 10, 14, 13, 1, 8, 5, 17, 16, 4, 13, 6, 12, 1, 5, 15, 2, 9, 19, 12, 17, 11, 5, 14, 10, 18, 4, 6, 16, 20, 19, 10, 13, 4, 6, 15, 22, 11, 3, 5
(list;
graph;
refs;
listen;
history;
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internal format)
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OFFSET
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1,3
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REFERENCES
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A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
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FORMULA
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EXAMPLE
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The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
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MAPLE
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# alternative
end proc:
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MATHEMATICA
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pmax = 1000; x[p_] := Module[{x, y}, x /. ToRules[Reduce[0 <= x <= y && x^2 + y^2 == p, {x, y}, Integers]]]; For[n=1; p=2, p<pmax, p = NextPrime[p], If[Mod[p, 4] == 1 || Mod[p, 4] == 2, a[n] = x[p]; Print["a(", n, ") = ", a[n]]; n++]]; Array[a, n-1] (* Jean-François Alcover, Feb 26 2016 *)
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PROG
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(PARI) f(p)=my(s=lift(sqrt(Mod(-1, p))), x=p, t); if(s>p/2, s=p-s); while(s^2>p, t=s; s=x%s; x=t); s
forprime(p=2, 1e3, if(p%4-3, print1(sqrtint(p-f(p)^2)", ")))
(PARI) do(p)=qfbsolve(Qfb(1, 0, 1), p)[2]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A002365
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Numbers y such that p^2 = x^2 + y^2, 0 < x < y, p = A002144(n).
(Formerly M3430 N1391)
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+10
15
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4, 12, 15, 21, 35, 40, 45, 60, 55, 80, 72, 99, 91, 112, 105, 140, 132, 165, 180, 168, 195, 221, 208, 209, 255, 260, 252, 231, 285, 312, 308, 288, 299, 272, 275, 340, 325, 399, 391, 420, 408, 351, 425, 380, 459, 440, 420, 532, 520, 575, 465, 551, 612, 608, 609
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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REFERENCES
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A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
3^2 + 4^2 = 5^2, giving x=3, y=4, p=5 and we have the first terms of A002366, the present sequence and A002144.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A070079
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a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).
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+10
14
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3, 5, 15, 21, 35, 9, 45, 11, 55, 39, 65, 99, 91, 15, 105, 51, 85, 165, 19, 95, 195, 221, 105, 209, 255, 69, 115, 231, 285, 25, 75, 175, 299, 225, 275, 189, 325, 399, 391, 29, 145, 351, 425, 261, 459, 279, 341, 165, 231, 575, 465, 551, 35, 105, 609, 315, 589, 385, 675
(list;
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OFFSET
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1,1
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COMMENTS
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Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2.
Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - Lekraj Beedassy, Jul 22 2005
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LINKS
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FORMULA
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EXAMPLE
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The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
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MATHEMATICA
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pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015
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STATUS
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approved
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4, 12, 8, 20, 12, 40, 28, 60, 48, 80, 72, 20, 60, 112, 88, 140, 132, 52, 180, 168, 28, 60, 208, 120, 32, 260, 252, 160, 68, 312, 308, 288, 180, 272, 252, 340, 228, 40, 120, 420, 408, 280, 168, 380, 220, 440, 420, 532, 520, 48, 368, 240, 612, 608, 200, 572, 300, 552, 52, 260
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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LINKS
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EXAMPLE
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The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A002366
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Numbers x such that x^2 + y^2 = p^2 = A002144(n)^2, x < y.
(Formerly M2442 N0970)
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+10
13
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3, 5, 8, 20, 12, 9, 28, 11, 48, 39, 65, 20, 60, 15, 88, 51, 85, 52, 19, 95, 28, 60, 105, 120, 32, 69, 115, 160, 68, 25, 75, 175, 180, 225, 252, 189, 228, 40, 120, 29, 145, 280, 168, 261, 220, 279, 341, 165, 231, 48, 368, 240, 35, 105, 200, 315, 300, 385, 52, 260, 259
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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REFERENCES
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A. J. C. Cunningham, Quadratic and Linear Tables. Hodgson, London, 1927, pp. 77-79.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Corrected definition to require p=A002144(n), which defines the order of the terms. - M. F. Hasler, Feb 24 2009
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STATUS
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approved
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25, 169, 289, 841, 1369, 1681, 2809, 3721, 5329, 7921, 9409, 10201, 11881, 12769, 18769, 22201, 24649, 29929, 32761, 37249, 38809, 52441, 54289, 58081, 66049, 72361, 76729, 78961, 85849, 97969, 100489, 113569, 121801, 124609, 139129
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OFFSET
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1,1
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COMMENTS
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a(n) is the sum of two positive squares in only one way. See the Dickson reference, (B) p. 227.
a(n) is the hypotenuse of two and only two right triangles with integral legs (modulo leg exchange). See the Dickson reference, (A) p. 227.
In 1640 Fermat generalized the 3,4,5 triangle with the theorem: A prime of the form 4n+1 is the hypotenuse of one and only one right triangle with integral arms. The square of a prime of the form 4n+1 is the hypotenuse of two and only two... The cube of three and only three...
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publ. No. 256, Vol II, Washington, DC, 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.
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LINKS
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FORMULA
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Product_{n>=1} (1 + 1/a(n)) = A243380
Product_{n>=1} (1 - 1/a(n)) = A088539. (End)
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EXAMPLE
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a(7) = 2809 is the hypotenuse of triangles 1241, 2520, 2809 and 1484, 2385, 2809, and only of these.
a(7) = 53^2 = 2809 = 45^2 + (4*7)^2, and this is the only way. - Wolfdieter Lang, Jan 13 2015
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MATHEMATICA
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PROG
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(PARI) fermat(n) = { for(x=1, n, y=4*x+1; if(isprime(y), print1(y^2" ")) ) }
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Edited: Name changed, part of old name as comment. Comments added and changed. Dickson reference added. - Wolfdieter Lang, Jan 13 2015
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STATUS
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approved
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A277557
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The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.
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+10
4
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8, 18, 19, 32, 33, 34, 50, 52, 53, 72, 73, 74, 75, 76, 98, 99, 100, 101, 102, 103, 128, 131, 133, 134, 162, 163, 164, 165, 166, 167, 168, 169, 200, 201, 202, 203, 204, 205, 206, 207, 208, 242, 244, 247, 248, 250, 251, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 338
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OFFSET
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1,1
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COMMENTS
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The mapping of the ordered pair (x,y) to an integer uses Cantor's pairing function to generate the integer as (x+y)(x+y+1)/2+y. Also for every ordered pair (x,y) such that 0 < x < y, gcd(x,y)=1 and x+y odd, there exists a primitive Pythagorean triple (PPT) (a, b, c) such that a = y^2-x^2, b = 2xy, c = x^2+y^2. Therefore each term in the sequence represents a unique PPT.
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LINKS
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EXAMPLE
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a(5)=33 because the ordered pair (2,5) maps to 33 by Cantor's pairing function (see below) and is the 5th such occurrence. Also x=2, y=5 generates a PPT with sides (21,20,29).
Note: Cantor's pairing function is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2, thus A001477(2,5) = 5 + (2+5)*(2+5+1)/2 = 33. - Antti Karttunen, Nov 02 2016
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MATHEMATICA
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Cantor[{i_, j_}] := (i+j)(i+j+1)/2+j; getparts[n_] := Reverse@Select[Reverse[IntegerPartitions[n, {2}], 2], GCD@@#==1 &]; pairs=Flatten[Table[getparts[2n+1], {n, 1, 20}], 1]; Table[Cantor[pairs[[n]]], {n, 1, Length[pairs]}]
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PROG
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(Scheme, with Antti Karttunen's IntSeq-library)
(define A277557 (MATCHING-POS 1 1 (lambda (n) (let ((x (A025581 n)) (y (A002262 n))) (and (not (zero? x)) (< x y) (odd? (+ x y)) (= 1 (gcd x y))))))) ;; Antti Karttunen, Nov 02 2016
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CROSSREFS
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Cf. A020882 (is obtained when A048147(a(n)) is sorted into ascending order), A008846 (same with duplicates removed).
Cf. A020887, A020888, A120427, A024362, A024406, A046079, A046087, A070151, A156678, A156679, A156680, A156683, A156685, A222946, A278147.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A145010
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a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 (mod 4).
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+10
3
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6, 30, 60, 210, 210, 180, 630, 330, 1320, 1560, 2340, 990, 2730, 840, 4620, 3570, 5610, 4290, 1710, 7980, 2730, 6630, 10920, 12540, 4080, 8970, 14490, 18480, 9690, 3900, 11550, 25200, 26910, 30600, 34650, 32130, 37050, 7980, 23460, 6090, 29580, 49140, 35700
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OFFSET
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1,1
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COMMENTS
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Pythagorean primes, i.e., primes of the form p = 4k+1 = A002144(n), have exactly one representation as sum of two squares: A002144(n) = x^2+y^2 = A002330(n+1)^2+A002331(n+1)^2. The corresponding (primitive) integer-sided right triangle with sides { 2xy, |x^2-y^2| } = { A002365(n), A002366(n) } has area xy|x^2-y^2| = a(n). For n>1 this is a(n) = 30*A068386(n).
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LINKS
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FORMULA
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EXAMPLE
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The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
p a b t_1 c d t_2 t_3 t_4
---------------------------------
5 1 2 1 3 4 4 3 6
13 2 3 3 5 12 12 5 30
17 1 4 2 8 15 8 15 60
29 2 5 5 20 21 20 21 210
37 1 6 3 12 35 12 35 210
41 4 5 10 9 40 40 9 180
53 2 7 7 28 45 28 45 630
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MATHEMATICA
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Reap[For[p = 2, p < 500, p = NextPrime[p], If[Mod[p, 4] == 1, area = x*y/2 /. ToRules[Reduce[0 < x <= y && p^2 == x^2 + y^2, {x, y}, Integers]]; Sow[area]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2015 *)
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PROG
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(PARI) forprime(p=1, 499, p%4==1 | next; t=[p, lift(-sqrt(Mod(-1, p)))]; while(t[1]^2>p, t=[t[2], t[1]%t[2]]); print1(t[1]*t[2]*(t[1]^2-t[2]^2)", "))
(PARI) {Q=Qfb(1, 0, 1); forprime(p=1, 499, p%4==1|next; t=qfbsolve(Q, p); print1(t[1]*t[2]*(t[1]^2-t[2]^2)", "))} \\ David Broadhurst
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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