Wikipedia:Reference desk/Archives/Mathematics/2016 November 12

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November 12

Integer polynomial solution

Reference for finding integer solutions for y=ax^2+bx+c. — Preceding unsigned comment added by 2600:1004:B126:216C:49EA:4FE0:89F3:BE6C (talk) 18:06, 12 November 2016 (UTC)

Please make the question a bit more explicit. Are you looking for a,b,c,x,y such that all five are integers, or integers x,y that fit given a,b,c, or what? —Tamfang (talk) 18:20, 12 November 2016 (UTC)

As usual, a b c are known. Then, x y are unknown. Want integer x y solutions. — Preceding unsigned comment added by 2600:1004:B126:216C:49EA:4FE0:89F3:BE6C (talk) 18:45, 12 November 2016 (UTC)

If a, b, and c are integers, a trivial way to algorithmically generate the solutions is to plug in all integer values of x and get the resulting value for y. If a, b, and c are rational, this can be extended to that case by first multiplying both sides by the least common denominator of a, b, and c, and then only keeping the values that are divided by the common denominator. The problem is much harder if a, b, and c are arbitrary real numbers; however, if a, b, and c are algebraically independent, then there can be no solution. This is an example of a Diophantine equation.--Jasper Deng (talk) 18:56, 12 November 2016 (UTC)

You can find all the solutions where y is set to any integer value by using the quadratic formula. You simply subtract the y value from both sides (from the c term of the right side) and find the solutions. They may be 1 or 2 solutions, which may be integers, real numbers, or even imaginary/complex. So, just pick the integer solutions out and discard the rest.

You can also set x = 0 and then y will be equal to c. StuRat (talk) 19:52, 12 November 2016 (UTC)

Except, that's easier said than done if we admit arbitrary real numbers for a, b, and c rather than rational numbers.--Jasper Deng (talk) 19:55, 12 November 2016 (UTC)

I assume you mean using the quadratic formula with a computer to generate a list. There we would have the problem of distinguishing between non-integer solutions and integer solutions which appear to be non-integers because of round-off error in the computer. Defining some arbitrarily small cut-of, like within one trillionth of an integer, as being considered to be an integer, would help, but we still might get some actual non-integers that this method considers to be integers. StuRat (talk) 20:02, 12 November 2016 (UTC)

parallel - hexagon?

Is there a name for all hexagons where the opposite sides (1,4), (2,5), (3,6) are parallel to each other? For a square that would be a parallelogram.Naraht (talk) 22:24, 12 November 2016 (UTC)

They are parallelogons, although that is the name of a larger class of shapes including parallelograms.--JohnBlackburne^words_deeds 22:28, 12 November 2016 (UTC)

But the article Parallelogon says that "opposite sides must be equal in length". So not all of the OP's hexagons are parallelogons. Loraof (talk) 14:36, 13 November 2016 (UTC)

Agreed with Loraof. I'm looking for a name that includes the hexagon gotten from snipping "relatively" tiny unequal size equilateral triangles off of the corners of an equilateral triangle.Naraht (talk) 16:32, 13 November 2016 (UTC)

Not an answer to your question, but you might be interested in rectilinear polygons,which have opposite sides parallel (along with a requirement on angles). Loraof (talk) 20:43, 13 November 2016 (UTC)