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Browse College Number Theory
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Testing for primality.

Stirling's Formula [05/17/2003]
I am having trouble finding an algorithm to solve the following problem: What's the least number for x such that x! >= 3^x is true?

Summing Activity Leads to a Mean of e [04/01/2005]
I asked my students to keep adding random integers from 1 to 100 until the sum exceeded 100. We then found the average number of terms added. The answer seems to be e. Why? The more we do it, the closer we get.

Sum of 1/Sqrt(i) [11/20/2000]
What is the formula for the sum of 1/sqrt(i) for i = 1 to n? Can you show me the proof by induction?

Sum of a Product of Floors of Square Roots ... Approximately [03/30/2016]
An adult seeks a closed-form solution for a complicated sum. Doctor Vogler unpacks it enough to convince himself that the sum might lack a closed form — and so sets about deriving an approximation.

Sum of Distinct Fibonacci Numbers [05/06/2001]
How do you show that every positive integer is a sum of distinct terms of the Fibonacci sequence?

Sum of First N Positive Integers Making a Perfect Square [02/28/2006]
For what integer values of n and k does 1 + 2 + 3 + ... + n = k^2?

Sum of Quadratic Residues [07/22/1999]
Given a prime number p = 4k+3. If d = (sum quadratic nonresidues) - (sum quadratic residues) in (0,p) can we prove that d is greater than 0 for all such p? Can we prove that d goes to infinity when p goes to infinity?

Sum of Sequence: a, b, a+b, a+2b, 2a+3b... [01/26/2003]
Find the sum of the first 30 terms of the sequence: 1, 5, 6, 11, 17, 28... if the 30th term is 2888956 and the 31st term is 4674429.

The Sum of Squares and Its Square Integer Quotient [07/19/2015]
A teen struggles to prove that dividing (a^2 + b^2) by (ab + 1) yields the square of an integer. Exploiting some symmetries, Doctor Vogler breaks the proof down into two claims.

Sum of Two Squares [05/26/2003]
Can you generate the sequence [400, 399, 393, 392, 384, 375, 360, 356, 337, 329, 311, 300]?

Sums of Consecutive Numbers [06/20/2002]
In what way(s) can 1000 be expressed as the sum of consecutive numbers?

System-Level Programming and Base 2 [05/03/2001]
In computer programming, I have a result that contains several values, always a power of 2 (2^2, 2^3, 2^4). If my value is 2^3, 2^4, 2^6 304, how can I tell if 2^3 exists in 304?

Testing primality of 32-bit numbers [11/18/1994]
What is the best (fastest) way to test if an arbitrary 32-bit number is prime?

Testing Prime Numbers [05/12/2003]
Besides the Sieve of Eratosthenes, what other methods can be used to determine all prime numbers within a given range? Is there a more efficient method?

Transfinite Arithmetic [10/28/1997]
What is transfinite arithmetic? I pretty much know what it means, but I am having trouble applying it to aleph-null.

Transfinite Numbers [11/07/1997]
I know that Georg Cantor discovered transfinite numbers, but what are they?

Triangle Perimeters [12/15/1998]
How many triangles with integer sides have a given perimeter? How does the triangle inequality enter into the proof?

Triangular Numbers [04/03/1997]
How do I prove that there are an infinite number of triangular numbers that are equal to a square number?

Tribonacci Numbers [11/11/2000]
Is there an implicit formula to calculate the nth Tribonacci number? Also, is there a formula to find the sum of the first n Tribonacci numbers?

Unique Decomposition of Pythagorean Primes [05/19/2002]
Is it true that a Pythagorean prime (i.e., a natural prime that can be expressed as a sum of squares of two integers) can be expressed as a sum of two squares in one and only one way?

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 Find All Solutions to the Diophantine Equation Ny + 1 = x^2 [12/13/2008]
I want to find all the solutions to the Diophantine equation Ny + 1 = x^2 without resorting to an exhaustive attack.

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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