I think that

Problem statement: Rice

Search for positive integer solutions to the equation x^{2}+y^{2}+z^{2}=w^{2}. Are there any? 0Eliminate any duplicates and solutions that are not primitive. What can you say about these solutions? In particular, which variables must be odd and which must be even?

(4n)

(4n+1)

(4n+2)

(4n+3)

So this means that odd squares have remainders of 1 when divided by 4, and the sum of two of them must have remainder of 2 when divided by 4. But even numbers have remainders of 0 when divided by 4. Therefore the sum of two ODD squares and one EVEN square can't be equal to an EVEN square.

Here the initial report was given by

Problem statement: Wheat

We looked at the equations x^{2}+y^{2}=z^{2}and x^{2}+y^{2}=2z^{2}and x^{2}+y^{2}=3z^{2}. The number of positive integer solutions seems to change based on the multiplier of z^{2}. Begin to check systematically. That is, look at x^{2}+y^{2}=Qz^{2}when Q=4, Q=5, and so on. For some Q's, the equation has solutions, and for other Q's it doesn't seem to have solutions. Accumulate some evidence. You may then want to consult theThe On-Line Encyclopedia of Integer Sequencesand see if you can get some insight.

Jointly Mr. Rowland and I noted a wonderful and difficult to state fact: suppose we have some sort of equation which involves integers. There may or may not be integer solutions to this equation. There is no general method ("decision procedure" or algorithm) which tells if there are integer solutions. This fact is difficult to state precisely and it is quite difficult to understand. But it is true.

Here I think the report was given by

Problem statement: Corn

There certainly seem to be lots of Pythagorean triples. What if we start putting extra conditions on them?

yclose toz

Suppose we want positive integer solutions to x^{2}+y^{2}=z^{2}but look for y's which are close to the corresponding z's. You can search for ...

Solutions with y=z-1;

solutions with y=z-2;

solutions with y=z-3;

etc.

xclose toy

Suppose we want positive integer solutions to x^{2}+y^{2}=z^{2}but look for x's which are close to the corresponding y's. You can search for

Solutions with x=y+1;

solutions with x=y+2;

solutions with x=y+3;

etc.

Almost isoceles: so there are integers x and z with x

There turn out to be many, many long thin Pythagorean triangles. That
is, x^{2}+y^{2}=(x+1)^{2} has many
solutions. This equation (square things out) is just
y^{2}=2x+1. And, yes, there are many many odd numbers which
are squares.

I think this group was the first to report, and I think

Problem statement: Barley

Suppose we have a rectangular box with positive integer side lengths x, y, and z. The sides of this box have many different diagonals but they only come in 3 different lengths, say a, b, and c. A drawing of such a box is shown to the right. Can all of the six values (x, y, z, a, b, and c) be positive integers? So we want positive integer solutions to these equations:

x Solutions to this system are called^{2}+y^{2}=a^{2}; y^{2}+z^{2}=b^{2}; z^{2}+x^{2}=c^{2}.Euler bricks. Please search for solutions.

CommentIf we also want the box's main diagonal (the "space diagonal" connecting the most distant corners) to be an integer -- that is, x^{2}+y^{2}+z^{2}=d^{2}where d is an integer -- then no solution is known. It's not known if there isany solution, even one not yet found!

**Further comments** So Mr. Rowland tried this search in `Mathematica`, which is another sort of general
purpose symbolic/numerical/graphical program. There the search took
about 1 second, but also he tried `Maple`
and his version of a similar program took about half a second. This
seemed very different from what Mr. Hard had reported. Indeed,
Mr. Hard told us that his program had *not* done the pruning
above, so maybe `Maple` and `Mathematica` are not so different from `Matlab`. Still the `Java` implementation is better, which is not
surprising if you know about the designs and purposes of all of these
languages. This is still all interesting if one is a fanatic about
efficiency. And efficiency matters, because it determines whether
certain computations are truly realistic or merely hypothetical!

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