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Face-diagonal lengths of Euler bricks.
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125, 157, 244, 250, 267, 281, 314, 348, 365, 373, 375, 471, 488, 500, 534, 562, 625, 628, 696, 707, 725, 730, 732, 746, 750, 773, 785, 801, 808, 825, 843, 875, 942, 976, 979, 1000, 1037, 1044, 1068, 1095, 1099, 1119, 1124, 1125, 1193, 1220, 1250, 1256, 1335
COMMENTS
Euler bricks are cuboids all of whose edges and face-diagonals are integers.
It is not known whether any Euler brick with space-diagonals that are integers exists.
825 is the only integer common to the sets of edge lengths and of face-diagonal lengths <= 1000 for Euler bricks.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.
P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.
FORMULA
Integer edges a > b > c such that integer face-diagonals are d(a,b) = sqrt(a^2 + b^2), d(a,c) = sqrt(a^2 + c^2), d(b,c) = sqrt(b^2 + c^2).
EXAMPLE
For n=1, the edges (a,b,c) are (240,117,44) and the face-diagonals (d(a,b),d(a,c),d(b,c)) are (267,244,125).
Note the pleasing factorizations of the edge-lengths of this least Euler brick: 240 = 15*4^2; 117 = 13*3^2; 44 = 11*2^2.
Pythagorean Threesomes: triples of natural numbers defining the six legs of three Pythagorean triangles.
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44, 117, 240, 240, 252, 275, 88, 234, 480, 85, 132, 720, 160, 231, 792, 132, 351, 720, 480, 504, 550, 176, 468, 960, 170, 264, 1440, 220, 585, 1200, 720, 756, 825, 320, 462, 1584, 264, 702, 1440, 308, 819, 1680, 255, 396, 2160, 960, 1008, 1100, 352, 936, 1920, 480, 693, 2376, 396, 1053, 2160, 429, 880, 2340, 340, 528, 2880
COMMENTS
The sequence is sorted by increasing sums of triples and secondly by increasing order of first term.
The three numbers in a Pythagorean Threesome define the lengths of three sides of a tetrahedron with all integer length edges and one right angle vertex.
The sequence was calculated for the science fiction novel "The Fifth Jack" by Andreas Boe, Amazon books, 2014.
I do not have that book, but this sequence is closely related to (and may be an erroneous version of) A268396. - Arkadiusz Wesolowski, Feb 03 2016
FORMULA
x,y,sqrt(x^2+y^2) y,z,sqrt(y^2+z^2) z,x,sqrt(z^2+x^2)
EXAMPLE
(44,117,240) sqrt(44^2+117^2)=125 sqrt(117^2+240^2)=267 sqrt(240^2+44^2)=244
Sides of Pythagorean cuboids: triples (a, b, c) that are integral length sides of a rectangular cuboid for which the three face diagonals x, y, z also have integral length.
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44, 117, 240, 240, 252, 275, 88, 234, 480, 85, 132, 720, 160, 231, 792, 132, 351, 720, 140, 480, 693, 480, 504, 550, 176, 468, 960, 170, 264, 1440, 220, 585, 1200, 720, 756, 825, 320, 462, 1584, 264, 702, 1440, 280, 960, 1386, 187, 1020, 1584, 308, 819, 1680
COMMENTS
Sides in increasing order of perimeter (a+b+c), where a < b < c.
A triple (a, b, c) of integers belongs to this sequence if and only if all of the numbers sqrt(a^2 + b^2), sqrt(b^2 + c^2), and sqrt(a^2 + c^2) are also integers.
Consider the set S(n) = {a(3*n-2), a(3*n-1), a(3*n)}. Then:
- at least one number in the set is divisible by 5
- at least one number in the set is divisible by 9
- at least one number in the set is divisible by 11
- at least one number in the set is divisible by 16
- at least two numbers in the set are divisible by 3
- at least two numbers in the set are divisible by 4.
The list of "Sides of ..." is A195816, while this sequence lists "Triples...", i.e., (a(3n-2), a(3n-1), a(3n)) = ( A031175(k), A031174(k), A031173(k)) for some k, n >= 1. (The order is not the same as for A031173 etc, e.g., the 5th through 8th triple have decreasing largest sides.) Also, in A031173, A031174, A031175 and others, the side naming convention is a > b > c, the opposite of here. - M. F. Hasler, Oct 11 2018
REFERENCES
Eli Maor, The Pythagorean Theorem: A 4,000-Year History, 2007, Princeton University Press, p. 134.
CROSSREFS
See A245616 for a very similar sequence.
Occurrences of edge-lengths of Euler bricks in every 100 consecutive integers.
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1
3, 8, 9, 8, 9, 9, 6, 9, 10, 8, 7, 9, 6, 8, 7, 8, 11, 6, 7, 8, 9, 8, 7, 6, 8, 10, 6, 6, 6, 8, 8, 8, 8, 9, 6, 9, 7, 6, 7, 8, 8, 9, 7, 11, 7, 8, 5, 9, 8, 9, 9, 7, 6, 7, 9, 6, 7, 9, 7, 8, 10, 5, 9, 7, 7, 7, 7, 6, 9, 9, 6, 8, 7, 9, 8, 6, 9, 5, 9, 9, 8, 6, 6, 7, 7
COMMENTS
Distribution of edge-length occurrences for Euler bricks is remarkably near-uniform.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.
P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.
EXAMPLE
For n=1 (i.e., the integers 1..100), there are only 3 possible edge-lengths for Euler bricks: 44, 85, 88.
PROG
(Sage)
def a(n):
ans = set()
for x in range(100*(n-1)+1, 100*n+1):
divs = Integer(x^2).divisors()
for d in divs:
if (d <= x^2/d): continue
if (d-x^2/d)%2==0:
y = (d-x^2/d)/2
for e in divs:
if (e <= x^2/e): continue
if (e-x^2/e)%2==0:
z = (e-x^2/e)/2
if (y^2+z^2).is_square(): ans.add(x)
Lengths of largest face diagonal in primitive Euler bricks or Pythagorean cuboids: possible values of max(d, e, f) for solutions to a^2 + b^2 = d^2, a^2 + c^2 = e^2, b^2 + c^2 = f^2 in coprime positive integers a, b, c, d, e, f.
+10
1
267, 373, 732, 825, 843, 1595, 1884, 2500, 2775, 3725, 3883, 6380, 6409, 8140, 8579, 9188, 9272, 9512, 11764, 12125, 13123, 14547, 14681, 14701, 19572, 20503, 20652, 24695, 25121, 25724, 29307, 30032, 30695, 31080, 32595, 34484, 37104, 37895, 38201, 38965
COMMENTS
These are the values obtained as sqrt( A031173(n)^2 + A031174(n)^2), sorted by size (n = 3 yields 843, n = 4 yields 732) and duplicates removed: The first duplicate is 71402500^2 = A031173(1428)^2 + A031174(1428)^2 = A031173(1626)^2 + A031174(1626)^2, there is no other among the first 3500 terms. This considers only the face diagonals, not the space diagonals.
See the main entry A031173 for links, cross-references, and further comments.
PROG
(PARI) A306120=Set(vector(1000, n, sqrtint( A031173(n)^2+ A031174(n)^2)))[1..-100] \\ Discard the last 100 values, which may have holes. This is empirical: better find the smallest sqrtint( A031173(n)^2+ A031174(n)^2) with n > 1000 not in the set, and discard all elements larger than that.
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