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Browse College Number Theory
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Selected answers to common questions:
    Testing for primality.

Explaining the Euclidean Algorithm [10/27/1998]
In the Euclidean Algorithm (or the Division Algorithm), why is the last divisor the greatest common factor?

Exploring Fermat's Little Theorem [02/08/2001]
Besides n is prime, are there any other sufficient conditions for Fermat's Little Theorem, a^(n-1) = 1 mod n with gcd(a,n) = 1?

Exponential Diophantine Equation [06/24/2005]
Find three integers a,b,c > 1 such that a^a * b^b = c^c.

Exponential Diophantine Equations [06/26/2005]
I'm wondering how to solve the following Diophantine equation. I need to get as many non-trivial solutions as possible: a^A = bB + c^C as a,b,c are given and relatively prime.

Exponential Series Proof [05/05/2001]
Given e^x greater than or equal to 1 + x for all real values of x,and that (1+1)(1+(1/2))(1+(1/3))...(1+(1/n)) = n+1, prove that e^(1+(1/2)+ (1/3)+...+(1/n)) is greater than n. Also, find a value of n for which 1=(1/2)+(1/3)+...+(1/n) is greater than 100.

Factorials' Final Figure [01/25/2017]
A recreational mathematician seeks ways to quickly determine the digit in the ones place of large factorials. By exploiting divisibility and carefully applying several simplifications, Doctor Vogler obliges, then extends the method to produce the last two digits of a given factorial.

Factoring Algorithms [08/11/2004]
I'm wondering if there is any sort of algorithm for taking a very large number and factoring it?

Factoring Large Numbers [10/26/1998]
Can you give me an algorithm for factoring large numbers? What about the Pollard Rho Factoring Algorithm?

Factorization Algorithms [02/22/1999]
The most efficient classical algorithm available for factoring a very large number (say, at least, 100 digits) into primes.

Farey Series [10/21/2002]
For three successive terms in a Farey's series, say a/b, c/d, e/f, how can we prove independently that c/d = (a+e)/(b+f) and ad-bc = -1 ?

Fastest Primality Test? [12/21/2005]
I found an article on the Internet discussing a new test to determine if a given number is prime or composite. Is it the fastest test?

Fermat Number Proof [01/30/2001]
Prove that if n is greater than 0, then the Fermat number 2^2^n + 1 is of the form 9k-1 or 9k-4. Prove that n and 2^2^n + 1 are relatively prime for every n greater than 0.

Fermat Numbers [01/25/2003]
How can we prove that any two Fermat Numbers are coprime? Are there infinitely many Fermat Numbers? How do we know that each Fermat Number will have a prime divisor?

Fermat's Last Theorem and Euler [11/29/2004]
I remember reading that Euler easily showed the case x^3 + y^3 = z^3 had no integer solutions, but I haven't been able to find any site that explains how to show this.

Fermat's Last Theorem for n = 3 [12/14/1998]
What is the proof for Fermat's Last Theorem where n = 3? Who is given credit for the first proof for this case?

Fermat's Last Theorem for n = 3 and 4 [12/16/2004]
A list of some web resources for discussion of Fermat's Last Theorem for n = 3 and 4.

Fermat's Little Theorem [09/02/2000]
Can you help me prove Fermat's Little Theorem, that the expression n^p-n, where p is an arbitrary prime and n is a positive integer, is always divisible by p?

Fermat's Little Theorem: A Special Case [06/26/2001]
Show that n^7-n is divisible by 7.

Fermat's Theorem (Not the Last) [07/14/1999]
Prove that if n is a whole number and p any prime, then n multiplied by itself p times minus n is divisible by p.

Fibonacci Formula Inductive Proof [11/05/1997]
I must prove by induction that F(n) = (PHI^n - (1 - PHI)^n) / sqrt5...

Fibonacci or Lucas Number [02/19/2003]
How do I know that any number x is a Fibonacci or Lucas number?

Fibonacci Sequence Property [11/29/2001]
I have to prove that in the Fibonacci sequence, F(k) is a divisor of F(nk), where n is a natural number (so, F(nk) = A*F(k) where A is a natural number).

Fibonacci's Liber Quadratorum - Proposition 18 [04/07/2002]
Prove by contradiction that if any two positive integers have an even sum, then the ratio of their sum to their difference will not be the same as the ratio of the larger number to the smaller.

Find a, b, c, Such That a! b! = a! + b! + c! [12/09/2003]
Find all triples of nonnegative integers a, b, c such that a! b! = a! + b! + c!

Find a Counterexample [10/19/2002]
Prove or disprove by counterexample that if p is a prime number, 2^p-1 must also be prime.

Find All Pairs of Integer Solutions to a^(2b) = b^a [02/18/2005]
A discussion between Doctor Vogler and a student about the problem of finding all pairs of positive integers (a,b) that satisfy the equation a^(2b) = b^a.

Finding a Desired Perfect Cube [04/26/2007]
What is the smallest positive cube that ends with the digits 2007?

Finding a Series Given the Sum [09/27/1999]
How can I find all series of consecutive integers whose sum is a given value x?

Finding Catalan Numbers [12/15/1999]
What are Catalan numbers and what applications do we have for them?

Finding Digits in Specific Decimal Places of Large Numbers [03/16/2006]
Find the digits immediately before and after the decimal point in (root2 + root3)^2000.

Finding Formulas for Number Sequences [11/22/1997]
My question is about trying to find a formula between numbers.

Finding GCD of Complex Numbers with Euclidean Algorithm [10/11/2004]
I would like to calculate GCD(135 - 14i, 155 + 34i) via the Euclidean algorithm, but I don't know how to do that with complex numbers.

Finding Integer Pairs Whose Product Consists Only of 1's and 0's [10/06/2004]
Given the base-10 representation of any integer a, does there exist a non-zero integer b such that the base-10 representation of the product ab contains only ones and zeros?

Finding Integer Solutions of x^3 - y^2 = 2 [06/01/2000]
How can I find all integer solutions of the equation x^3 - y^2 = 2 and prove that they are the only solutions?

Finding Integer Solutions to a^b = b^a [02/02/2005]
Find all positive integers a and b such that a^b = b^a, and prove that you have found them all.

Finding Integer Solutions to Four-Variable Equation [06/10/2007]
Find four different integers a, b, c and d such that a^2 + b^2 + c^2 + d^2 = a*b*c*d.

Finding Primitive Solutions [10/11/2000]
How do I find all primitive solutions (x,y,z) to x^2 + 5*y^2 = z^2? Once I have them, how can I generate other integer solutions?

Finding the Exponent with a Modulus [05/24/2003]
I am trying to work out k in the following question: 23^k = 201545 (mod 900001).

Finding the Last Digits of a Large Exponential [01/01/2005]
What are the last five digits of 7777777^7777777?

Finding the Last Few Non-Zero Digits of Large Factorials [10/08/2007]
Is there a way to find the last x non-zero digits of N! without having to do out all the multiplications?

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