See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse High School Number Theory
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Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Pythagorean Triples [05/22/1999]
What is the general formula for all sides of any triple?
- Pythagorean Triples [05/31/1999]
Is there a procedure for finding Pythagorean triples?
- Pythagorean Triples [5/18/1995]
How can the relation between Pythagorean triples be expressed as a
- Pythagorean Triples Divisible by 5 [11/17/2000]
Do all right triangles with integer side lengths have a side with a
length divisible by 5?
- Pythagorean Triple with 71 [12/07/1997]
Is there a Pythagorean triple that contains the number 71?
- Quadratic Residues [06/30/1998]
I need a fundamental explanation of the concept of quadratic residues.
- Quadratic Residues and Sums of Squares [10/28/1998]
In one of the lemmas in number theory, if p is an odd prime number, then
there exist x, y such that x^2+y^2+1=kp...
- A Quartic Diophantine Equation: 10657 + 11579x^2 + x^4 = y^2 [12/29/2008]
Doctor Vogler helps a student look for integer solutions to a quartic
polynomial by noticing a difference of squares in its coefficients and
factoring its constant term.
- Ramsey's Theorem and Infinite Sequence [06/01/1999]
Ramsey's Theorem applied to divisibility in infinite sequences.
- Rational and Irrational Numbers: Multiplication, Division [10/15/2001]
I would like the rules explained for: irrational * irrational; rational *
- Rationalizing a Denominator with Multiple Cube Roots [04/22/2011]
A student of field theory wonders how to remove the cube roots from the denominator
of 1/(a + b*CBRT(q) + c*CBRT(q)^2). Building on the conjugacy of square roots,
Doctor Vogler writes out the required conjugates.
- Real and Rational Numbers [02/27/2001]
How can I show that the number of rational numbers between 0 and 1 is the
same as the number of natural numbers (considering the ordering of
fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)?
- Real Numbers [08/08/1997]
What exactly is a real number?
- Reasoning about Integers [10/06/2004]
When positive integers p and q are divided by an even positive integer
t, they have remainders 2 and t/2, respectively. What is the remainder
when the product pq is divided by t?
- Reciprocals of Integers Greater Than 1 as Sum of a Series [07/01/2004]
Show that the reciprocal of every integer greater than 1 is the sum of
a finite number of consecutive terms of the series 1/[j(j + 1)].
- Rectangular Solids from Blocks [09/25/1998]
How many rectangular solids can be made from "n" cube-shaped blocks?
- Recurrence Relation for a Pell Equation [11/09/1999]
Can you help me find a recurrence relation for generating solutions to
the Pell equation x^2 - 5y^2 = 1?
- Relationship Between GCF and LCM [05/22/2002]
What is the exact relationship between the gcf or gcd and the lcm of
- Relatively Prime [10/07/1999]
What does the term relatively prime mean, and how can you determine if
two numbers are relative primes?
- Relatively Prime Pythagorean Triples [09/13/1997]
Questions about Pythagorean triples.
- Remainders, Pigeons, and Pigeonholes [03/26/2003]
Given 17 integers, prove that it is always possible to select 5 of the
17 whose sum is divisible by 5.
- Remainder when Dividing Large Numbers [04/17/2001]
How can I find the remainder when (12371^56 + 34)^28 is divided by 111?
- Repeating Decimals [04/28/1999]
I am interested in finding longer repeating groups in number tails of
- Repeating Decimals - Rational or Irrational? [09/11/2001]
Are 0.252252225... and 0.125126127... rational or irrational?
- Repeating Digits of Fractions [04/28/1999]
Do you know any theorems relating to the length of the repeating portion
of the decimal representation of fractions?
- Reversal of Age Digits Every Eleven Years [11/06/2007]
Every 11 years, my age is the exact reverse of my mother's age. When I
was 13 she was 31, when I was 24 she was 42, and so on. Why does this
- Reversed Digits Theorem [06/24/1999]
For a positive integer abc..., if (abc...)^n = xyz... and if
(a+b+c+...)^n = x+y+z+..., how can I prove that (...cba)^n = ...zyx?
- Reverse Modulus Operator [10/09/2001]
Is there an operator that would return 2 when we we do 6 * 0, * being
this new operator?
- Roots of ax^f2 bx+c = 0 [05/22/1997]
Prove that if a,b,c are odd integers, then the roots of ax^2 bx+c=0 are
- Rounding Negative Numbers [05/23/2007]
I was wondering what -0.5 is rounded to the nearest integer?
- RSA Encryption [04/25/2002]
Decrypt the encrypted message in ciphertext C to find the original
plaintext, a string of English letters.
- Running Time of an Insertion Sort [08/06/1999]
How can I write and solve a recurrence formula for the running time of an
insertion sort? Which is better, an insertion sort or a merge-sort?
- Search for the Largest Prime [08/01/2000]
What is the largest finite number that has a practical use in some branch
of mathematics or science? What is the largest prime number known?
- Second-Degree Two-Variable Diophantine Equation [04/12/2001]
Solve Ax^2+Bxy+Cy^2+Dx+Ey+F = 0 where B^2-4AC=k^2 for some integer k.
- Second-Order Linear Recurrences [06/08/2001]
Three problems involving recurrence equations.
- Second Order Recurrence with Non-Constant Coefficients [05/27/2005]
I'm trying to find a closed form solution of a second order recurrence
relation with no constant coefficients, specifically: u(n+2) =
2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help?
- Sequence of Integers [08/12/2008]
Find all functions f such that for each n in Z+ we have f(n) > 1 and
f(n + 3)f(n + 2) = f(n + 1) + f(n) + 18.
- Set Theory and GCD and Divisibility [01/27/2003]
If 1 <= a <= n and 1 <= b <= n and ab <= n, and if a divides n and b
divides n, does that mean that ab divides n given that GCD(a,b) = 1 ?
- Show 2^(N-1) Congruent to 1(mod N) [02/25/2003]
I need to show that if N = 2^p - 1, p prime, then 2^(N-1) is congruent
to 1(mod N).
- Showing a Diophantine Equation Has No Solutions [07/30/2008]
Do there exist positive integers m and n such that m^3 = 3n^2 + 3n + 7?