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A057102 Congrua (possible solutions to the congruum problem): numbers n such that there are integers x, y and z with n = x^2-y^2 = z^2-x^2. 13
24, 96, 120, 240, 336, 384, 480, 720, 840, 960, 1320, 1344, 1536, 1920, 1944, 2016, 2184, 2520, 2880, 3360, 3696, 3840, 3960, 4896, 5280, 5376, 5544, 6144, 6240, 6840, 6864, 7680, 7776, 8064, 8736, 9240, 9360, 9720, 10080, 10296, 10920, 11520, 12144 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Each congruum is a multiple of 24; it cannot be a square.

Numbers of the form (4(x^3y-xy^3) (where x,y are integrs and x>=y). Squares of these numbers are of the form N^4-K^2 (where N belongs to A135786 and K to A135789 or A135790). Proof uses identity: (4(x^3y-xy^3))^2=(x^2+y^2)^4-(x^4 - 6x^2 y^2 + y^4)^2 - Artur Jasinski, Nov 29 2007, Nov 14 2008

LINKS

Table of n, a(n) for n=1..43.

Wikipedia, Congruum

Eric Weisstein's World of Mathematics, Congruum.

FORMULA

a(n) = 4 * A073120(n). - Max Alekseyev, Nov 14 2008

EXAMPLE

a(9)=840 since 840=29^2-1^2=41^2-29^2 (indeed also 840=37^2-23^2=47^2-37^2)

MATHEMATICA

a = {}; Do[Do[w = 4x^3y - 4x y^3; If[w > 0 && w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a] - Artur Jasinski, Nov 29 2007

CROSSREFS

Cf. A004431 for possible values of x in definition. Cf. A057103, A055096 for triangles of all congrua and values of x.

Cf. A073120, A135789, A135786.

Sequence in context: A208984 A103251 A198387 * A057103 A055669 A209432

Adjacent sequences:  A057099 A057100 A057101 * A057103 A057104 A057105

KEYWORD

nonn,changed

AUTHOR

Henry Bottomley, Aug 02 2000

STATUS

approved

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Last modified December 18 10:28 EST 2014. Contains 252140 sequences.