A215365 Primitive integer length of the side of an origin-centered square that contains inside its boundary a point with integer coordinates that is an integer distance from three of the four corners.
52, 700, 740, 996, 3364, 6240, 7800, 8400, 10952, 11184, 11352, 11492, 11484, 13156, 20280, 20988, 21320, 22472, 26180, 26588, 28168, 34500, 39988, 40680, 43700, 44944, 45976, 49500, 58956, 70448, 77500, 90168, 103896, 105468, 106200, 115752, 118636, 124620, 129000
Offset: 1
Keywords
Examples
With n = side length, we find an a,b such that a^2 + b^2 = d1^2, a^2 + (n-b)^2 = d2^2, b^2 + (n-a)^2 = d3^2, (n-a)^2 + (n-b)^2 = d4^2 is true in integers for three of these four equations. n = 52 is the first, with a=7 and b=24.
Links
- Yasushi Ieno, Other special cases of a square problem, arXiv:2111.02888 [math.GM], 2021.
- Yang Ji, Several special cases of a square problem, arXiv:2105.05250 [math.GM], 2021.
- G. Shute and K. L. Yocom, Problem 966. Seven integral distances, Math. Mag. 50, 166 (1977).
- UnsolvedProblems Web Site, Rational Distance
Crossrefs
Cf. A260549.
Programs
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PARI
has(n)=for(a=1,n-1,for(b=a,n-1, if(issquare(norml2([a,b])) + issquare(norml2([n-a,b])) + issquare(norml2([a,n-b])) + issquare(norml2([n-a,n-b])) > 2, return(1)))); 0 is(n)=sumdiv(n,d,has(d))==1 \\ Charles R Greathouse IV, Jul 28 2015
Extensions
Data corrected and name improved by Mark Underwood, Jul 28 2015
a(7)-a(39) from Giovanni Resta, Jul 29 2015
Comments