can someone ask me a tough Algebra question or equation?

Question

also provide the answer - i need to ask my students an algebra question or equation for extra credit.

Answer 1

Fine an interger solution to a^n + b^n = c^n where n > 3.

This one is very hard.

Answer 2

Here are three that have appeared recently.
I gave the second one to my students as a bonus recently.

1. Suppose x>2 and y >2. Can x^y + y^x ever be a
power of another integer?

2. Let a, b, c be the sides of a triangle, with c
the longest side.
We all know that if a²+b² = c² the triangle is
a right triangle. There are infinitely many triangles
with integer sides which satisfy this equation.
For example, 3,4,5 , 5,12,13, 7,24,25, etc.
Now suppose that c² = a²+b²+ab.
Using the law of cosines, it's not hard to show
that the largest angle of this triangle is 120 degrees.
Problem: Find 3 whole numbers, all larger than 1,
which satisfy this equation.

3). Here's one we've been kicking around for
years, with no success.
Let n be a whole number
greater than 1 (in base 10) satisfying
1). n is odd
2). n is a square
3). The only digits of n are 0 and 1.
Find the smallest such n or show no such n exists.
This problem is said to have originated in a
middle-school math. book.
We have also shown by direct computation
that no such n exists if n < 10^36.
Here is a tantalising near miss:
375501² = 141001001001.

Sorry, I have no answers to any of these.
The first and last of these are open questions,
I think!

Answer 3

For an extra-credit question, huh? How about this:

A magician picks a member of the audience to think of any three digit number, and write it on a chalkboard. The magician then asks the participant to reverse the digits to make a new three-digit number, and finally subtract the smaller from the bigger (e.g. 721 - 127 = 594). Then the person is asked to take this new number, reverse the digits, and add the results together (e.g. 594 + 495=1089). At this point the magician pulls out a sealed envelope containing the correct answer of 1089. Prove that the answer is always 1089, regardless of what number you start with (HINT: if "ABC" is the three digit number, it can be written as 100A + 10B + C).

Or some trick questions from Martin Gardner:

A harmonica costs a dollar more than a pencil. Together, they cost $1.10. How much does each item cost?
[If they answer 10 cents for the pencil and $1 for the harmonica, they're wrong!]

A farmer came to town to sell watermelons. He sold half of what he had plus half a watermelon, and found that he had one whole melon left. How many did he bring to town?

Answer 4

Proof that a^3+b^3 = c^3 has no natural solution.

Answer 5

Find an interger solution to a^n + b^n = c^n where n > 3.

good luck........ and no i didnt copy!

Answer 6

Find a solution to x^(x^(x^...))) = 2

Similarly, find a solution to x = sqrt(x + sqrt(x + sqrt(x + ...)))