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LaGrange's Theorem [02/24/2001]

Please explain LaGrange's Theorem on the number of roots of a polynomial.

Large Prime Numbers [12/17/1997]

Is there an algorithm to determine whether a very large number is prime?

Large Prime Numbers [01/13/2009]

What is the largest prime number less than which all primes are known?

Largest Mersenne Prime [1/19/1995]

What is the largest currently known Mersenne prime?

Last Eight Digits of Z [01/18/2002]

Consider the recursive function f(n)=13^(f(n-1)), where f(0)=13^13. A=13^13, B=13^(13^13), etc. A=f(0), B=f(A), C=f(B),... Z=f(Y). What are the last 8 digits of Z?

Last Four Digits of the Fibonacci Numbers [05/06/2001]

Show that there is a number ending with four zeros in the Fibonacci sequence; prove that the Fibonacci sequence has a cycle for the last four digits with a length of 15,000.

Linear Congruences of Gaussian Integers [04/11/2003]

When does the linear congruence zx congruent to 1 (mod m), for z, x, and m all Gaussian integers, have a solution? Also, when do we say that two Gaussian integers are relatively prime?

Linear Diophantine Equation [02/05/2003]

Given positive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5,....., prove that a = b.

Linear Proof [11/07/2001]

We say that f is linear provided that for every x, y in its domain, f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R (the set of real numbers), then f is defined by f(x) = cx for some c belong to R.

Linear Recurrance Relations [08/10/2004]

Is there a general approach to taking a pattern that is defined recursively and finding an explicit definition for it?

List of Perfect Numbers [3/24/1995]

Do you know any Perfect Numbers besides 6, 28, and 496? Do you know where I can get a list of Perfect Numbers?

Mathematical Induction [01/28/2002]

Use Mathemetical Induction to prove that any postage of at least 8 cents can be obtained using 3- and 5-cent stamps.

Mathematica's PrimitiveRoot[] Function [11/14/2002]

My current project is to develop a fast primitive root finder.

Minimizing the Number of Stamps Needed for Postage [11/20/2007]

A student wants to write a computer program to find the minimum number of stamps needed to create a desired postage given the various stamp values that are currently available. Doctor Tom explains why that sort of problem presents a very difficult challenge.

Missile Launch Code [08/03/2003]

What kind of information could you give all 10 people such that if any 3 of them were to get together, they would be able to launch the missiles, but if there were only 2 of them, the information would be insufficient to figure out the code?

Mod, Inverses, and Number Theory [11/27/2001]

I already have values for e, p and q, and need to find a value for d in: d = e^-1 mod (p-1)(q-1).

Modular Form Ingredients in Fermat's Last Theorem [12/13/2000]

The proof of Fermat's Last Theorem shows how the L-series of elliptic curves and the M-series of modular forms correspond, and how modular forms are composed of different "ingredients." Can you provide a description of the ingredients of a modular form?

Modular Square Roots [03/24/2006]

Is there an algorithm for finding modular square roots where the modulus is either a square or an odd composite?

Modulus Congruence Proof [04/18/2001]

How can I prove 2^(3n+2)+21n = 4 mod (49)?

Modulus Proof [04/16/2001]

Can you please show me why m^(2^n) = 1 mod(2^(n+2)) when m is an odd integer?

More on Order of Operations [02/13/2000]

I have found contradictory information on the precedence of the multiplication and division operations. Is there a universally accepted rule for the order of these operations?

Multiplication of Integers Modulo (2^16 + 1) [10/18/2002]

Prove that 2^16 * 2^15 mod (2^16 + 1) = 2^15 + 1.

Multiplicative Groups of Order (p-1) [05/12/2000]

What is the proof that primitive roots for multiplicative groups of order (p-1), where p is prime, exist? Is there an algorithm for finding them?

No Solution: y^2 = x^3 + 7 [03/17/2003]

Show that y^2 = x^3 + 7 has no integer solution.

A Number Digits Puzzle [02/23/2001]

How can I determine all positive integers with the property that they are one more than the sum of the squares of their digits?

Numbers in the Fibonacci Sequence [07/19/2001]

How can I show that there is a number in the Fibonacci sequence that ends in 999999999999 ? For what numbers n is there a number in the Fibonacci sequence that ends in n of 9 ?

Number Theory and Modular Arithmetic [01/27/2006]

If the order of an element mod(p) = n, is it true that the order of this element mod(p^k) = n * p^{k-1}, where p is odd prime and k > 1?

Number Theory - Perfect Square [5/26/1996]

Find all the possible values of n...

Number Theory: Sum of Cube [7/6/1996]

Generate the sum of cube for a given integer n...

Odd Numbers and Modulo 8 [09/06/2004]

Is it true that any odd number greater than 1, multiplied by itself, is congruent to 1 modulo 8? If so, why?

The Official Euclidean Algorithm [11/16/2000]

Can you state briefly the "official" Euclidean Algorithm?

Only Two Abelian Groups [02/25/2003]

Show that any group with order p^2, p is a prime, is Abelian. Show that up to isomorphism that only two such groups exist.

Orbits of the Baker's Map Function [12/11/2000]

I'm investigating the number of orbits a particular input has for the Baker's map function... why are all the values with 1 orbit the primes of primitive root 2?

Overview of Riemann's Zeta Function and Prime Numbers [04/11/2006]

Can you please give an overview of the importance of the Zeta function and finding prime numbers? Why is the Zeta function such a hot topic in the field of looking for prime numbers?

Pairs of Integers [08/16/1997]

Show that there are infinitely many pairs of integers(x,y) such that x|y**2+m and y|x**2+m where m is any chosen integer; moreover gcd(x,y)=1.

Palindrome Problem [2/6/1995]

According to the "Reversing Number Algorithm," it's thought that most (perhaps all) numbers eventually wind up being palindromes. Have you seen any references about this problem?

Partitions and Products [01/02/2003]

What is the best way of dividing an integer into parts so that the product of the parts will be as large as possible? Is there a universal law that tells us what partition will produce the maximum product for any given number? And can such a law be proved?

Patterns in Repeating Decimals [08/06/2003]

Why do certain number sequences repeat in the decimal expansions of common fractions?

Perfect Numbers [06/15/2002]

Please show that any even perfect number ends in 6 or 8.

Perfect Square [10/26/2001]

If a and b are positive integers such that (1+ab) divides (a^2+b^2), show that the integer (a^2+b^2)/(1+ab) must be a perfect square.

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