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Using Binomial Expansion to Evaluate [2 + sqrt(3)]^50 [11/29/2006]

I've used a computer to evaluate [2 + sqrt(3)]^50 and the answer is extremely close to being an integer. I've tried various expansions of the expression to try and determine why it's so close to an integer, but haven't gotten anywhere. Do you have any idea why?

Using Elliptical Curves to Solve an Arithmetic Sequence [05/02/2006]

Find a three-term arithmetic sequence of rational numbers such that the product of the three terms is 11.

Using Gaussian Integers to Solve a Diophantine Equation [05/30/2008]

Find an integer solution to x^2 + y^2 = 26,819,945 without trying all values for x or y. You are allowed to use the factorization of 26,819,945 if necessary.

Using Modular Arithemtic to Find a Remainder [08/06/2008]

Could you devise a simple rule to find the remainder of a number when it's divided by 13?

Using Modular Arithmetic to Test Divisibility of Large Numbers [08/30/2008]

Prove that 55^62 - 2*13^62 + 41^62 is divisible by 182.

Using Two Irrationals to Generate All Positive Integers [10/03/2003]

If a and b are positive irrational numbers such that 1/a + 1/b = 1, then every positive integer can be uniquely expressed as either floor (ka) or floor(kb), where k is a positive integer.

Vandermonde's Convolution [05/23/2002]

Prove that (nC0)^2 + (nC1)^2 +... + (nCn)^2 = (2n)!/(n!)^2.

Was Euler wrong? 2*Pi=0? [03/13/2002]

While I was surfing the Internet, I found a site with an interesting proof that shows that 2*Pi = 0 by using Euler's famous equation...

What are the Factors of 33550336? [4/9/1995]

Do you know the factors for the Perfect Number, 33550336?

What Makes Polynomials Relatively Prime? [11/20/2007]

Why are polynomials whose only common factors are constants considered 'relatively prime'? Why are the common constants not considered? For example, 3x + 6 and 3x^2 + 12 are considered relatively prime even though they have a common constant factor of 3.

When is the Sum of n Square Numbers Also a Perfect Square? [10/10/2005]

The formula P(n) = n(n + 1)(2n + 1)/6 generates the sum of the first n square numbers, so that P(3) = 14 = 1 + 4 + 9. I found that P(24) = 4900, which is a square number. Are there other cases where the sum of n squares is also a perfect square?

Why 2 + 2 = 4 [10/27/1995]

This is an original question I have been asked by a student: Why does 2 + 2 = 4?

Why does 2+2 = 4? [6/4/1996]

You say that the hard thing to show is that 1+1 = 2, but that 4 is just another name for 1+1+1+1. Isn't this a little incongruent?

x^2 + y^2 Is Composite? [01/30/2003]

Prove or give a counterexample: For all integers, if x + y is composite and x - y is composite, then x^2 + y^2 is composite.

Zero of a Monic Polynomial [02/05/2003]

Show that a zero of a monic polynomial is irrational or is an integer.

Z Transforms and the Fibonacci Sequence [04/20/1998]

Can you suggest an example of using Z transforms to derive the equation of a Fibonacci number?

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