A002330 -id:A002330 - cp's OEIS frontend

A002340 Numbers x such that p = x^2 - 5y^2, where p == 0, 1, or 4 (mod 5).

5, 4, 8, 7, 6, 11, 8, 9, 14, 18, 13, 11, 17, 16, 12, 13, 14, 28, 19, 14, 18, 16, 27, 22, 31, 16, 17, 26, 19, 34, 24, 23, 22, 28, 37, 41, 27, 32, 21, 26, 22, 23, 31, 22, 44, 48, 23, 29, 39, 43, 24, 38, 51, 32, 26, 31, 28, 29, 36, 41, 27, 58, 28

Offset: 1

N. J. A. Sloane

nonn

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]

Cf. A002341.

Corrected and extended by Sean A. Irvine, Oct 21 2013

Offset corrected by Mohammed Yaseen, Jul 24 2023

A002341 Numbers y such that p = x^2 - 5y^2, where p = 0, 1, or 4 (mod 5).

Original entry on oeis.org

2, 1, 3, 2, 1, 4, 1, 2, 5, 7, 4, 2, 6, 5, 1, 2, 3, 11, 6, 1, 5, 3, 10, 7, 12, 1, 2, 9, 4, 13, 7, 6, 5, 9, 14, 16, 8, 11, 2, 7, 3, 4, 10, 1, 17, 19, 2, 8, 14, 16, 1, 13, 20, 9, 3, 8, 5, 6, 11, 14

Offset: 1

N. J. A. Sloane

nonn

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]

Cf. A002340.

Offset corrected by Mohammed Yaseen, Jul 24 2023

A002342 Least positive integer x such that p = (x^2 - 5*y^2)/4 where p is the n-th odd prime such that 5 is a square mod p.

Original entry on oeis.org

5, 7, 9, 11, 12, 13, 16, 17, 17, 19, 19, 22, 21, 23, 24, 26, 27, 29, 27, 28, 29, 32, 31, 31, 33, 32, 34, 33, 37, 37, 37, 39, 41, 39, 41, 43, 41, 41, 42, 43, 44, 46, 43, 44, 47, 49, 46, 47, 47, 49, 48, 49, 53, 51, 52, 53, 56, 57, 53, 53, 54, 59, 56, 57, 58, 59, 59, 57, 58, 61

Offset: 1

N. J. A. Sloane

nonn

The n-th odd prime for which 5 is a square mod p is A038872(n).

			5 = (5^2 - 5*1^2)/4 so a(1)=5;
11 = (7^2 - 5*1^2)/4 so a(2)=7.

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]

Cf. A002343, A038872.

a(n)=local(y,p); if(n<1,0,p=0; y=2; until(p>=n,y++; if(issquare(5+O(prime(y))),p++)); p=prime(y); y=0; until(issquare(4*p+5*y^2),y++); sqrtint(4*p+5*y^2))

A002343 Least positive integer y such that p = (x^2 - 5*y^2)/4 where p is the n-th odd prime such that 5 is a square mod p.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 1, 3, 1, 4, 1, 1, 2, 4, 5, 5, 1, 2, 3, 6, 3, 1, 5, 2, 4, 1, 7, 5, 3, 5, 7, 1, 5, 7, 3, 1, 4, 5, 6, 8, 1, 2, 7, 9, 4, 5, 3, 5, 2, 1, 9, 5, 6, 7, 10, 11, 3, 1, 4, 11, 6, 7, 8, 9, 7, 1, 4, 9, 5, 3, 8, 13, 3, 1, 4, 11, 1, 8, 2, 9, 10, 11, 13, 14, 7, 4, 5, 11, 7, 2, 10, 11, 15, 5, 9

Offset: 1

N. J. A. Sloane

nonn

The n-th odd prime for which 5 is a square mod p is A038872(n).

			5 = (5^2 - 5*1^2)/4 so a(1)=1;
11 = (7^2 - 5*1^2)/4 so a(2)=1.

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]

Cf. A002342.

Cf. A002342, A038872.

a(n)=local(y,p); if(n<1,0,p=0; y=1; until(p>=n,y++; if(issquare(5+O(prime(y))),p++)); p=prime(y); y=0; until(issquare(4*p+5*y^2),y++); y)

A068386 One-thirtieth the area of the unique Pythagorean triangle whose hypotenuse is A002144(n), the n-th prime of the form 4k+1.

Original entry on oeis.org

1, 2, 7, 7, 6, 21, 11, 44, 52, 78, 33, 91, 28, 154, 119, 187, 143, 57, 266, 91, 221, 364, 418, 136, 299, 483, 616, 323, 130, 385, 840, 897, 1020, 1155, 1071, 1235, 266, 782, 203, 986, 1638, 1190, 1653, 1683, 2046, 2387, 1463, 2002, 460, 2852, 2204, 357

Offset: 2

Lekraj Beedassy, Mar 08 2002

Every such prime p has a unique representation as p = r^2 + s^2 with 1 <= r < s. The corresponding right triangle has legs of lengths s^2 - r^2 and 2rs and area rs(s^2 - r^2). For p > 5, this is divisible by 30.
Calling A002330(n) and A002331(n) respectively u and v, we have a(n) = u*v*(u-v)*(u+v), for n > 1. - Lekraj Beedassy, Mar 12 2002
The corresponding Pythagorean triple (A, B, C) with A^2 = B^2 + C^2, (A > B > C) is given by {A002144(n), A002365(n), A002366(n)}, so that a(n) = B*C/(2*30) = A002365(n)*A002366(n)/60. - Lekraj Beedassy, Oct 27 2003

			The 7th prime of the form 4k+1 is 53 = 2^2 + 7^2. So the right triangle has sides 7^2 - 2^2 = 45, 2*2*7 = 28 and 53. Its area is 1/2 * 45 * 28 = 630, so a(7) = 630/30 = 21.

Cf. A008846, A002144.

a30[p_] := For[r=1, True, r++, If[IntegerQ[s=Sqrt[p-r^2]], Return[r s(s^2-r^2)/30]]]; a30/@Select[Prime/@Range[4, 150], Mod[ #, 4]==1&]
areat[p_]:=Module[{c=Flatten[PowersRepresentations[p,2,2]],a,b},a= First[c];b= Last[c];((b^2-a^2)(2a b))/2]; areat[#]/30&/@Select[Prime[ Range[4,200]],IntegerQ[(#-1)/4]&] (* Harvey P. Dale, Jun 21 2011 *)

Edited by Dean Hickerson, Mar 14 2002

A244290 Smallest prime a(n) = x^2 + y^2 such that c^2 + d^2 = A002313(n) and cx + dy = 1, where c,d,x,y are integers.

Original entry on oeis.org

5, 2, 2, 53, 5, 173, 2, 17, 2, 29, 13, 5, 1697, 53, 2, 73, 13, 5, 37, 2, 389, 733, 2753, 89, 17, 1093, 773, 13, 397, 1789, 2, 41, 821, 53, 5, 29, 193, 281, 6257, 173, 2, 149, 593, 701, 5, 1289, 157, 5, 7993, 13, 2213, 449, 877, 2, 61, 37, 389, 17, 5, 24061

Offset: 1

Thomas Ordowski, Jun 27 2014

nonn

Let c^2 + d^2 = p be a prime, A002313(n). Then x^2 + y^2 = q is the smallest prime, a(n), such that cx + dy = 1 (Bézout's identity), where c,d,x,y are integers. We have pq = m^2 + 1 at m = cy - dx.
a(n) is the smallest prime q such that q*A002313(n)-1 is a square. - Thomas Ordowski, Sep 13 2015
Conjecture: a(n) < A002313(n)^2 for n > 1. - Thomas Ordowski, Dec 28 2017

			For prime 2 = 1^2 + 1^2 is 1*2 + 1*(-1) = 1 and 2^2 + (-1)^2 = 5 is prime, so a(1) = 5. For A002313(2) = 5 is vice versa so a(2) = 2.

Robert Israel, Table of n, a(n) for n = 1..2910

Cf. A002313, A002330, A002331.

N:= 10^6: # to get all a(n) before the first one > N
P:= select(isprime, [2,seq(4*i+1, i=1..floor((N-1)/4))]):
f:= proc(p) local i;
  for i from 1 to nops(P) do
   if issqr(p*P[i]-1) then return P[i] fi
od:
  -1
end proc:
for i from 1 to nops(P) do
  v:= f(P[i]);
if v = -1 then break fi;
A[i]:= v;
od:
seq(A[j],j=1..i-1); # Robert Israel, Sep 13 2015

\\ cs should contain terms from A002330
\\ ds should contain terms from A002331
a244290(cs, ds) = {
  vector(#cs, i,
    c=cs[i]; d=ds[i]; [u,v]=gcdext(c, d);
    x=u; y=v; while(!isprime(x^2+y^2), x+=d; y-=c); e=x^2+y^2;
    x=u; y=v; while(!isprime(x^2+y^2), x-=d; y+=c); f=x^2+y^2;
    min(e, f)
  )
} \\ Colin Barker, Jul 06 2014

More terms from Colin Barker, Jul 06 2014

A080680 Integer part of the square root of the n-th prime of the form 4k+1.

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 13, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29

Offset: 1

Reinhard Zumkeller, Mar 02 2003

nonn

a(n)^2 < A002330(n+2)^2 + A002331(n+2)^2 < a(n+1)^2.

Floor[Sqrt[#]]&/@Select[Prime[Range[300]],Mod[#,4]==1&] (* Harvey P. Dale, Jun 04 2023 *)

a(n) = A000196(A002144(n)).

A107961 Pythagorean semiprimes: products of two Pythagorean primes (A002313).

Original entry on oeis.org

4, 10, 25, 26, 34, 58, 65, 74, 82, 85, 106, 122, 145, 146, 169, 178, 185, 194, 202, 205, 218, 221, 226, 265, 274, 289, 298, 305, 314, 346, 362, 365, 377, 386, 394, 445, 458, 466, 481, 482, 485, 493, 505, 514, 533, 538, 545, 554, 562, 565, 586, 626, 629, 634

Offset: 1

Jonathan Vos Post, Jun 12 2005

Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique manner (up to the order of addends) in the form x^2 + y^2 for integer x and y iff p = 1 (mod 4) or p = 2 (which is a degenerate case with x = y = 1). The theorem was stated by Fermat, but the first published proof was by Euler.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 146-147 and 220-223, 1996.
Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 13 and 219, 1979.
Seroul, R. "Prime Number and Sum of Two Squares." Section 2.11 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 18-19, 2000.

Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002313, A002330, A002331.

{a(n)} = {p*q: p and q both elements of A002313} = {p*q: p and q both of form m^2 + n^2 for integers m, n}.

A383176 If p = A002313(n) is a prime such that p = x^2 + y^2, then a(n) is the largest integer k that satisfies x^2 + y^2 - kxy > 0.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 3, 2, 3, 2, 2, 10, 3, 2, 3, 2, 2, 6, 2, 2, 14, 7, 2, 4, 16, 2, 2, 3, 8, 2, 2, 2, 3, 2, 2, 2, 3, 20, 6, 2, 2, 3, 5, 2, 4, 2, 2, 2, 2, 24, 3, 5, 2, 2, 6, 2, 4, 2, 26, 5, 2, 13, 3, 2, 2, 2, 2, 5, 2, 3, 2, 7, 5, 2, 2, 2, 3, 2, 7, 5, 2, 2, 3

Offset: 1

Gonzalo Martínez, Apr 18 2025

nonn

Fermat's Christmas theorem states that if p = 2 or if p is congruent to 1 modulo 4 (A002313), then p is written as a sum of 2 squares uniquely. Thus, if A002313(n) = x^2 + y^2, for certain integers x and y, then a(n) is the largest integer k such that x^2 + y^2 - k*x*y > 0.

a(n) >= 2, for n > 1. If p > 2 and p = x^2 + y^2, since x != y, then it is satisfied that 0 < (x - y)^2 = x^2 + y^2 - 2x*y < x^2 + y^2 - x*y. The equality a(n) = 2 is given when |x - y| < phi*min{x, y}.

			Since A002313(8) = 53 and 53 = 2^2 + 7^2, we have that 53 - 3*2*7 > 0 and 53 - 4*2*7 < 0, then a(8) = 3.

Cf. A002313, A002330, A002331.

from itertools import count, islice
from sympy import isprime
from sympy.solvers.diophantine.diophantine import cornacchia
def A383176_gen(): # generator of terms
    yield 1
    for p in count(5,4):
        if isprime(p):
            for x,y in cornacchia(1,1,p):
                yield p//(x*y)
A383176_list = list(islice(A383176_gen(),30)) # Chai Wah Wu, Apr 26 2025

Definition clarified by Chai Wah Wu, Apr 26 2025

cp's OEIS Frontend

A002340 Numbers x such that p = x^2 - 5y^2, where p == 0, 1, or 4 (mod 5).

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A002341 Numbers y such that p = x^2 - 5y^2, where p = 0, 1, or 4 (mod 5).

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A002342 Least positive integer x such that p = (x^2 - 5*y^2)/4 where p is the n-th odd prime such that 5 is a square mod p.

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A002343 Least positive integer y such that p = (x^2 - 5*y^2)/4 where p is the n-th odd prime such that 5 is a square mod p.

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A068386 One-thirtieth the area of the unique Pythagorean triangle whose hypotenuse is A002144(n), the n-th prime of the form 4k+1.

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A244290 Smallest prime a(n) = x^2 + y^2 such that c^2 + d^2 = A002313(n) and c*x + d*y = 1, where c,d,x,y are integers.

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Maple

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A080680 Integer part of the square root of the n-th prime of the form 4k+1.

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Mathematica

Formula

A107961 Pythagorean semiprimes: products of two Pythagorean primes (A002313).

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Formula

A383176 If p = A002313(n) is a prime such that p = x^2 + y^2, then a(n) is the largest integer k that satisfies x^2 + y^2 - k*x*y > 0.

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A244290 Smallest prime a(n) = x^2 + y^2 such that c^2 + d^2 = A002313(n) and cx + dy = 1, where c,d,x,y are integers.

A383176 If p = A002313(n) is a prime such that p = x^2 + y^2, then a(n) is the largest integer k that satisfies x^2 + y^2 - kxy > 0.