A008846 -id:A008846 - cp's OEIS frontend

A156685 Number of primitive Pythagorean triples A^2 + B^2 = C^2 with 0 < A < B < C and gcd(A,B)=1 that have a hypotenuse C that is less than or equal to n.

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12

Offset: 1

Ant King, Feb 17 2009

D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = 1/(2*Pi) = 0.1591549...

			There is one primitive Pythagorean triple with a hypotenuse less than or equal to 7 -- (3,4,5) -- hence a(7)=1.
G.f. = x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + 2*x^14 + ...

Lehmer, Derrick Norman; Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4, (Oct. 1900), pp. 293-335.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
Ramin Takloo-Bighash, How many Pythagorean triples are there?, A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer, Cham, 2018, 211-226.

Cf. A008846, A020882, A024409, A024362, A224921.

a156685 n = a156685_list !! (n-1)
a156685_list = scanl1 (+) a024362_list  -- Reinhard Zumkeller, Dec 02 2012

RightTrianglePrimitiveHypotenuses[1]:=0;RightTrianglePrimitiveHypotenuses[n_Integer?Positive]:=Module[{f=Transpose[FactorInteger[n]],a,p,mod1posn},{p,a}=f;mod1=Select[p,Mod[ #,4]==1&];If[Length[a]>Length[mod1],0,2^(Length[mod1]-1)]];RightTrianglePrimitiveHypotenuses[ # ] &/@Range[75]//Accumulate

a(n)=sum(a=1,n-2,sum(b=a+1,sqrtint(n^2-a^2), gcd(a,b)==1 && issquare(a^2+b^2))) \\ Charles R Greathouse IV, Apr 29 2013

Essentially partial sums of A024362.

A164620 Primes p such that 1 +p*floor(p/2) is also prime.

Original entry on oeis.org

2, 5, 13, 17, 41, 61, 89, 97, 101, 113, 149, 173, 229, 241, 281, 317, 349, 353, 373, 397, 409, 421, 433, 461, 509, 521, 661, 673, 761, 853, 881, 937, 941, 1013, 1093, 1109, 1249, 1289, 1297, 1373, 1433, 1549, 1741, 1753, 1913, 2113, 2213, 2269, 2281, 2297

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

			p=2 qualifies since 2*1+1=3 is prime. p=5 qualifies since 5*2+1=11 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008846, A020882, A158708.

lst={};Do[p=Prime[n];If[PrimeQ[p*Floor[p/2]+1],AppendTo[lst,p]],{n,3*6!}]; lst
Select[Prime[Range[350]],PrimeQ[1+#*Floor[#/2]]&] (* Harvey P. Dale, Apr 07 2015 *)

Comments turned into examples by R. J. Mathar, Sep 17 2009

A277557 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.

Original entry on oeis.org

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

Offset: 1

Frank M Jackson, Oct 19 2016

nonn

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.
Numbers n for which 0 < A025581(n) < A002262(n) and A025581(n)+A002262(n) is odd, and gcd(A025581(n), A002262(n)) = 1. [The definition expressed with A-numbers.] - Antti Karttunen, Nov 02 2016
See also the triangle T(y, x) with the values for PPTs given in A278147. - Wolfdieter Lang, Nov 24 2016

			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

Wikipedia, Cantor's pairing function, and Pythagorean triple

Cf. A001477, A002262, A025581, A243808, A277632.
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.

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]}]

A072592 Even numbers with at least one prime factor of form 4*k+1.

Original entry on oeis.org

10, 20, 26, 30, 34, 40, 50, 52, 58, 60, 68, 70, 74, 78, 80, 82, 90, 100, 102, 104, 106, 110, 116, 120, 122, 130, 136, 140, 146, 148, 150, 156, 160, 164, 170, 174, 178, 180, 182, 190, 194, 200, 202, 204, 208, 210, 212, 218, 220, 222, 226, 230, 232, 234, 238

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

Conjecture: this is exactly the sequence whose terms are twice those of A009003. (This has been verified for all terms<=500.) Compare A009003. - John W. Layman, Mar 12 2008

The conjecture is true. See comments on A008846 and A004613. - Lambert Herrgesell (zero815(AT)googlemail.com), Apr 24 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005843, A002144, A072591, A009003, A008846, A004613.

opfQ[n_]:=Count[Transpose[FactorInteger[n]][[1]],?(IntegerQ[ (#-1)/4]&)]>0; Select[Range[2,250,2],opfQ] (* _Harvey P. Dale, Jun 02 2012 *)

from itertools import count, islice
from sympy import primefactors
def A072592_gen(startvalue=1): # generator of terms >= startvalue
    return (n<<1 for n in count(max(startvalue,1)) if any(p&3==1 for p in primefactors(n)))
A072592_list = list(islice(A072592_gen(),20)) # Chai Wah Wu, Jul 07 2024

A072591(a(n)) = 0.

A087937 Prime hypotenuse of primitive Pythagorean triangles with even short leg (or odd long leg).

Original entry on oeis.org

17, 29, 37, 53, 73, 101, 109, 137, 173, 197, 229, 241, 257, 281, 293, 349, 373, 397, 401, 409, 449, 457, 509, 577, 593, 601, 641, 661, 677, 701, 733, 757, 809, 829, 857, 877, 941, 977, 997, 1021, 1033, 1049, 1061, 1093, 1153, 1181, 1193, 1229, 1237, 1277

Offset: 1

Lekraj Beedassy, Oct 27 2003

nonn

Ordered intersection of A002144 and A081985.

These same triangles have the property that sqrt(h+c) = t, where h = a(n), c is the even short leg, t is an integer and the odd long leg is an odd multiple of t, but it appears there are no cases where that multiple is 1, in contrast with the similar property that applies to the triangles of A087937. Also see A008846. - Richard R. Forberg, May 11 2016

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A002144, A008846, A081985, A087937.

More terms from Ray Chandler, Oct 28 2003

A208853 Array of hypotenuses of primitive Pythagorean triangles when read by SW-NE diagonals.

Original entry on oeis.org

5, 13, 17, 29, 25, 37, 53, 41, 0, 65, 85, 65, 61, 73, 101, 125, 97, 85, 89, 109, 145, 173, 137, 0, 113, 0, 0, 197, 229, 185, 157, 145, 149, 169, 205, 257, 293, 241, 205, 185, 181, 193, 221, 265, 325, 365, 305, 0, 233, 221, 0

Offset: 1

Wolfdieter Lang, Mar 05 2012

All primitive Pythagorean triples (see the links) (a,b,c), with a odd, b even, hence c odd, are given by c=u^2 + v^2, with u odd, u=2*n+1, n>=1, v even, v=2*m, m>=1, and gcd(u,v)=1. The present array is c=c(n,m) = (2*n-1)^2 + (2*m)^2, if gcd(2*n-1,2*m)=1 and 0 otherwise. The corresponding triangle, read by SW-NE diagonals, is T(n,m):= c(n-m+1,m). The 0 entries indicate that there are only non-primitive triples for these n,m values. See the example section for the scaling factor g=gcd(u,v)^2 for such non-primitive triangles.
For the increasingly ordered c-values see A008846 (with multiplicity see A020882).
All primitive Pythagorean triples are given by
(a(n,m)=A208854(n,m), b(n,m)=A208855(n,m), c(n,m)), n>=1, m>=1. If this is (0,0,0) then no primitive triple exists for these n,m values. See the example section.
In the prime factorization of c(n,m) (which is odd) all prime factors are of the type 4*k+1 (see A002144). See the Niven-Zuckerman-Montgomery reference, Theorem 3.20, p. 164. For the general representation of positive integers as the sum of two squares see Theorem 2.15 by Fermat, p. 55. E.g.: c(5,2) = 85 = 5*17. c = 5*7^2 = 245 has a non-primitive solution 7^2*(1^2 + 2^2) = 7^2*c(1,1), therefore c(4,7)=0 in this array.
The triples with an even cathetus (b) and the hypotenuse (c) differing by 1 unit are (2*k+1, 4*T(k), 4*T(k)+1), k >= 1, with the triangular numbers A000217. The c values are given in A001844. E.g., (n,m)=(1,1), k=1. (3,4,5); (n,m)=(2,1), k=2, (5,12,13); (n,m)=(2,2), k=3, (7,24,25). See the example section for the table.
The triples with an odd cathetus (a) and the hypotenuse differing by 2 units are (4*k^2-1, 4*k, 4*k^2+1), k >= 1. These triples are given in (A000466(k), A008586(k), A053755(k)). E.g., (n,m)=(1,4), k=4, (63,16,65).
The triples with the catheti differing by one length unit are generated by a substitution rule for the (u,v) values starting with (1,1). See a Wolfdieter Lang comment on A001653 for this rule. - Wolfdieter Lang, Mar 08 2012

			Triangle T(n,m):
......m|  1     2     3     4    5    6     7     8    9    10 ...
......v|  2     4     6     8   10   12    14    16   18    20 ...
n,  u
1,  1     5
2,  3    13    17
3,  5    29    25    37
4,  7    53    41     0    65
5,  9    85    65    61    73  101
6, 11   125    97    85    89  109  145
7, 13   173   137     0   113    0    0   197
8, 15   229   185   157   145  149  169   205   257
9, 17   293   241   205   185  181  193   221   265  325
10,19   365   305     0   233  221    0     0   281    0   401
...
Array c(n,m):
......m|  1    2    3     4     5    6     7     8    9    10 ...
......v|  2    4    6     8    10   12    14    16   18    20 ...
n,  u
1,  1     5   17   37    65   101  145   197   257  325   401
2   3    13   25    0    73   109    0   205   265    0   409
3,  5    29   41   61    89     0  169   221   281  349     0
4,  7    53   65   85   113   149  193     0   305  373   449
5,  9    85   97    0   145   181    0   277   337    0   481
6, 11   125  137  157   185   221  265   317   377  445   521
7, 13   173  185  205   233   269  313   365   425  493   569
8, 15   229  241    0   289     0    0   421   481    0     0
9, 17   293  305  325   353   389  433   485   545  613   689
10,19   365  377  397   425   461  505   557   617  685   761
...
------------------------------------------------------------------
Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)):
......m|    1           2               3             4  ...
......v|    2           4               6             8  ...
n,  u
1,  1    (3,4,5)     (15,8,17)      (35,12,37)     (63,16,65)
2,  3    (5,12,13)   (7,24,25)       (0,0,0)       (55,48,73)
3,  5   (21,20,29)   (9,40,41)      (11,60,61)     (39,80,89)
4,  7   (45,28,53)   (33,56,65)     (13,84,85)    (15,112,113)
5,  9   (77,36,85)   (65,72,97)      (0,0,0)      (17,144,145)
6, 11  (117,44,125) (105,88,137)   (85,132,157)   (57,176,185)
7, 13  (165,52,173) (153,104,185)  (133,156,205) (105,208,233)
8, 15  (221,60,229) (209,120,241)    (0,0,0)     (161,240,289)
9, 17  (285,68,293) (273,136,305)  (253,204,325) (225,272,353)
10,19  (357,76,365) (345,152,377)  (325,228,397) (297,304,425)
...
Array continued:
Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)):
......m|     5            6               7           8  ...
......v|    10           12              14          16  ...
n,  u
1,  1   (99,20,101)  (143,24,145)  (195,28,197)  (255,32,257)
2   3   (91,60,109)     (0,0,0)    (187,84,205)  (247,96,265)
3,  5    (0,0,0)     (119,120,169) (171,140,221) (231,160,281)
4,  7  (51,140,149)  (95,168,193)     (0,0,0)    (207,224,305)
5,  9  (19,180,181)     (0,0,0)    (115,252,277) (175,288,337)
6, 11  (21,220,221)  (23,264,265)  (75,308,317)  (135,352,377)
7, 13  (69,260,269)  (25,312,313)  (27,364,365)  (87,416,425)
8, 15     (0,0,0)       (0,0,0)    (29,420,421)  (31,480,481)
9, 17  (189,340,389) (145,408,433) (93,476,485)  (33,544,545)
10,19  (261,380,461) (217,456,505) (165,532,557) (105,608,617)
...
(0,0,0) indicates that no primitive Pythagorean triangle exists for these (n,m) values. The corresponding scaled triples would be (a,b,c) = g*(a/g,b/g,c/g), with g=gcd(u,v)^2 for such non-primitive triangles. E.g., c(n,m) = c(5,3) = 0, (u,v) = (9,6), g = 3^2, (45,108,117) = 3^2*(45/9,108/9,117/9) = 9*(5,12,13). The scaling factor for the primitive triangle (5,12,13), tabulated for c(n,m)=(2,1), is here 9.

I. Niven, H. S. Zuckerman and H.L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley & Sons, New York, 1991

Ron Knott, Pythagorean Triples and Online Calculators.
E. S. Rowland, Primitive Solutions to x^2 + y^2 = z^2
Michael Somos, Table of primitive Pythagorean triples and related parameters
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A020882, A002144, A208854, A208855.

T(n,m) = c(n-m+1,m), n >= m >= 1, with c(n,m) := (2*n-1)^2 + (2*m)^2, if gcd(2*n-1, 2*m) = 1 and 0 otherwise.

A239381 a(0) = 3, the least length of a Primitive Pythagorean Triangle (PPT). a(n) is the least hypotenuse of a PPT which has a(n-1) as one of its legs.

Original entry on oeis.org

3, 5, 13, 85, 157, 12325, 90733, 2449525, 28455997, 295742792965, 171480834409967437, 656310093705697045, 1616599508725767821225590944157, 4461691012090851100342993272805, 115366949386695884000892071602798585632943213, 12002377162350258332845595301471273220420939451301220405

Offset: 0

Robert G. Wilson v, Mar 17 2014

nonn

a(0)=3 because A042965(3)=3 with comments.

If we relax the Primitive restriction, i.e., GCD(x,y,z) can exceed 1, then we have A018928.

			a(0)=3 by definition,
a(1)=5 because it is the hypotenuse of a 3-4-5 PPT,
a(2)=13 because it is the hypotenuse of a 5-12-13 PPT,
a(3)=85 because it is the hypotenuse of a 13-84-85 PPT,
a(4)=157 because it is the hypotenuse of a 85-132-157 PPT, 85 is also the leg of a 85-3612-3613 PPT but its hypotenuse is larger, etc.

Robert G. Wilson v, Table of n, a(n) for n = 0..22

Cf. A008846, A235598.

f[s_List] := Block[{x = s[[-1]]}, Append[s, Transpose[ Solve[x^2 + y^2 == z^2 && GCD[x, y, z] == 1 && y > 0 && z > 0, {y, z}, Integers]][[-1, 1, 2]]]]; lst = Nest[f, {3}, 15]

A376210 Numbers k for which among all possible Pythagorean triangles with the hypotenuse 4*k+1, the minimum of the lengths of the shorter legs is even.

Original entry on oeis.org

4, 7, 9, 13, 16, 18, 25, 27, 34, 43, 49, 57, 60, 64, 70, 73, 81, 87, 93, 99, 100, 102, 111, 112, 114, 121, 123, 127, 133, 144, 148, 150, 157, 160, 165, 169, 175, 183, 186, 189, 196, 202, 207, 211, 214, 219, 225, 235, 241, 244, 249, 255, 256, 258, 262, 265, 273

Offset: 1

Hugo Pfoertner, Sep 21 2024

nonn

			       Hypotenuses                A376210
       4k+1                       |  A376211
   k   A008846                    |  |  A376208
   |   |  Sorted legs [x,y] of    |  |  |  A375750
   |   |  Pythagorean triangles   |  |  |  |  A376209
   1   5  [3,4]                   .  X  .  X  .
   3  13  [5,12]                  .  X  .  X  .
   4  17  [8,15]                  X  .  X  .  .
   6  25  [7,24]                  .  X  .  X  .
   7  29  [20,21]                 X  .  X  .  .
   9  37  [12,35]                 X  .  X  .  .
  10  41  [9,40]                  .  X  .  X  .
  13  53  [28,45]                 X  .  X  .  .
  15  61  [11,60]                 .  X  .  X  .
  16  65  [16,63],[33,56],[39,52] X  .  X  X  X
  18  73  [48,55]                 X  .  X  .  .
  21  85  [13,84],[36,77],[51,68] .  X  X  X  X

({A087937}-1)/4 is a subsequence.
A094178 \ A376211.
Cf. A375750, A376208.

is_a376210_1(n,r=0) = my(c=4*n+1, q=qfbsolve(Qfb(1,0,1), c^2, 3), qd=#q); if(qd<2, 0, my(a=vecmin(abs(concat(q))[1..2*(qd-1)]), b=sqrtint(c^2-a^2)); a%2==r && gcd([a,b,c])==1)

A057229 a(n) = ab = xy with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.

Original entry on oeis.org

6, 30, 60, 84, 210, 210, 180, 630, 330, 504, 924, 1320, 546, 1386, 1560, 2340, 990, 2730, 840, 2574, 4620, 1224, 1716, 3570, 5610, 7140, 4290, 1710, 5016, 7956, 7980, 2730, 7854, 10374, 2310, 11970, 6630, 10920, 12540, 4080, 3036, 11856, 8970

Offset: 0

Naohiro Nomoto, Sep 19 2000

nonn

The quadratics in X, X^2 - S*X -+ P, where S=A020882(n), P=A057229(n) are each factorizable into two factors, all four being distinct: X^2 - S*X - P = (X - a)*(X + b). X^2 - S*X + P = (X - x)*(X - y). - Lekraj Beedassy, Apr 30 2004

Areas of primitive Pythagorean triangles sorted on hypotenuse A020882, then on perimeter A093507. - Lekraj Beedassy, Aug 18 2006

			E.g. a(1)=6=6*1=3*2, (6-1)=(3+2)=5=A020882(1), gcd(6,1)=gcd(3,2)=1

P. Yiu, Factorizable x^2 + px -+ q, Recreational Mathematics, pp. 58/360.

Cf. A020882, A008846, A024406, A024365.

A087938 Prime hypotenuse of primitive Pythagorean triangles with odd short leg (or even long leg).

Original entry on oeis.org

5, 13, 41, 61, 89, 97, 113, 149, 157, 181, 193, 233, 269, 277, 313, 317, 337, 353, 389, 421, 433, 461, 521, 541, 557, 569, 613, 617, 653, 673, 709, 761, 769, 773, 797, 821, 853, 881, 929, 937, 953, 1009, 1013, 1069, 1097, 1109, 1117, 1129, 1201, 1213, 1217

Offset: 1

Lekraj Beedassy, Oct 27 2003

nonn

Ordered intersection of A002144 and A081961.

These same triangles have the property that sqrt(h+b) = t, where h = a(n), b is the even long leg, t is an odd integer, and the odd short leg is an odd multiple of t, including some instances where that multiple is 1 (i.e., t equals the short leg). A similar property applies to the triangles of A087937. Also see A008846. - Richard R. Forberg, May 11 2016

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A002144, A008846, A081961, A087937.

Corrected and extended by Ray Chandler, Oct 28 2003

cp's OEIS Frontend

A156685 Number of primitive Pythagorean triples A^2 + B^2 = C^2 with 0 < A < B < C and gcd(A,B)=1 that have a hypotenuse C that is less than or equal to n.

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A164620 Primes p such that 1 +p*floor(p/2) is also prime.

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A277557 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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A072592 Even numbers with at least one prime factor of form 4*k+1.

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A087937 Prime hypotenuse of primitive Pythagorean triangles with even short leg (or odd long leg).

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A208853 Array of hypotenuses of primitive Pythagorean triangles when read by SW-NE diagonals.

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A239381 a(0) = 3, the least length of a Primitive Pythagorean Triangle (PPT). a(n) is the least hypotenuse of a PPT which has a(n-1) as one of its legs.

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A376210 Numbers k for which among all possible Pythagorean triangles with the hypotenuse 4*k+1, the minimum of the lengths of the shorter legs is even.

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A057229 a(n) = ab = xy with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.