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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 closedform 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?
 SystemLevel 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 32bit numbers [11/18/1994]

What is the best (fastest) way to test if an arbitrary 32bit 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 alephnull.
 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 threeterm 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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