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

1,1

Comments

No point with integer distance to all four corners is known.

The sequence only contains even values because an odd-sided square centered at the origin has corners with non-integer coordinates, which cannot be at integer distance from interior lattice points. If the square instead of being centered at the origin has a corner on the origin, then the resulting sequence is A260549. - Giovanni Resta, Jul 29 2015

Links

Table of n, a(n) for n=1..39.

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

Example

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.

Prog

(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

Crossrefs

Cf. A260549.

Sequence in context: A254137 A249712 A255945 * A339142 A264309 A160344

Adjacent sequences: A215362 A215363 A215364 * A215366 A215367 A215368

Keyword

nonn

Author

Mark Underwood, Aug 08 2012

Extensions

Data corrected and name improved by Mark Underwood, Jul 28 2015

a(7)-a(39) from Giovanni Resta, Jul 29 2015

Status

approved