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A197040
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Occurrences of edge-lengths of Euler bricks in every 100 consecutive integers.
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1
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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
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OFFSET
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1,1
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COMMENTS
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Distribution of edge-length occurrences for Euler bricks is remarkably near-uniform.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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For n=1 (i.e., the integers 1..100), there are only 3 possible edge-lengths for Euler bricks: 44, 85, 88.
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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