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Irrationality of e+pi and e*pi [09/24/2001]

I have read that it is unknown whether either E+Pi or E*Pi is an irrational number. How can we prove that at most one of the two numbers is rational?

Irrational Numbers x,y, x^y Rational? [09/28/2001]

Are there any irrational numbers x and y such that x^y is rational?

The Jumping Frog of Coordinate County [11/03/2009]

A frog starts at (0,0) and always jumps a distance of exactly 1 unit to a point with rational coordinates. Show that it is possible for the frog to reach the point (1/5,1/17) and not to reach (0,1/4).

Lagrange's Theorem [02/27/2001]

In your archives you show proofs of Lagrange's theorem that every positive integer can be expressed as the sum of four squares, but is there an algorithm for identifying which four squares?

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?

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