01:090:101:21, Experimental Math, the seventh meeting

Diary for 01:090:101:21, Experimental Math, fall 2008

The seventh meeting, 10/20/2008

Discussion
Here are notes about what I hope will be discussed during this meeting. Some other relevant links follow.

The home page of Eric Rowland who will present today's material.

Information about the Pythagorean Theorem and Pythagorean triples
Some history
More thorough (Wikipedia) history

The home page of Noam Elkies

Why Euler's conjecture is false:
On A⁴+B⁴+C⁴=D⁴ by Noam Elkies in Mathematics of Computation, 51 (October 1988, 825-835). This link should work from any Rutgers terminal.
Discussion of Elkies' paper by Mr. Rowland, prepared in December 2004.

Biography of Leonhard Euler (1707–1783

Some questions we will ask students to investigate.

Changes!!!
Following a student suggestion, the logical question round(z)=z was used in class instead of type(z,integer). I like the student suggestion better than what we had planned. It is simpler and more related to how most people think.

The procedures discussed
Here are the procedures used today, put on a web page so that you may lift and copy them into a Maple screen more easily. At the urging of Mr. Rowland who is a kind and interested instructor, these procedures are presented with line breaks which are supposed to help human beings understand them. Internally Maple doesn't care about presentation that much.

Finds and prints the integers in the range from 1 to n which are squares of other integers.

squares := proc(n)
local x;
    for x to n do if type(sqrt(x), integer) then print(x) end if end do
end proc;

Finds and prints all Pythagorean triples a, b, and c (solutions of a²+b²=c²) which have a and b between 1 and n.

pythag := proc(n)
local x, y;
    for x to n do for y to n do
            if type(sqrt(x^2 + y^2), integer) then
                print(x, y, sqrt(x^2 + y^2))
            end if
        end do
    end do
end proc;

Prints Pythagorean triples with a and b between 1 and n and with a≤b.

pythagB := proc(n)
local x, y;
    for x to n do for y from x to n do
            if type(sqrt(x^2 + y^2), integer) then
                print(x, y, sqrt(x^2 + y^2))
            end if
        end do
    end do
end proc;

Prints primitive Pythagorean triples with a and b between 1 and n and with a≤b. The word "primitive" means that a and b have no integer common factor.

pythagC := proc(n)
local x, y;
    for x to n do for y from x to n do
            if type(sqrt(x^2 + y^2), integer) and igcd(x, y) = 1 then
                print(x, y, sqrt(x^2 + y^2))
            end if
        end do
    end do
end proc;

Prints all integer solutions to the equation a²+b²=2c² with a and b between 1 and n with a≤b and so that a and b have no common integer factors.

pythag2 := proc(n)
local x, y;
    for x to n do for y from x to n do
            if type(sqrt(1/2*x^2 + 1/2*y^2), integer) and igcd(x, y) = 1
            then print(x, y, sqrt(1/2*x^2 + 1/2*y^2))
            end if
        end do
    end do
end proc;

Prints all integer solutions to the equation a²+b²=3c² with a and b between 1 and n with a≤b and so that a and b have no common integer factors.

pythag3 := proc(n)
local x, y;
    for x to n do for y from x to n do
            if type(sqrt(1/3*x^2 + 1/3*y^2), integer) and igcd(x, y) = 1
            then print(x, y, sqrt(1/3*x^2 + 1/3*y^2))
            end if
        end do
    end do
end proc;

Maintained by greenfie@math.rutgers.edu and last modified 10/12/2008.