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'?
- Prime Number Theorems [01/03/1999]
Can you explain the prime number theorem, Mersenne primes, the Lucas-
Lehmer test, and the Riemann Hypothesis?
- Prime Proofs [10/08/2002]
If a^(n) - 1 is prime, show that a=2 and that n is a prime. If a^(n) +
1 is a prime, show that a is even and that n is a power of 2.
- Primes and Repeating Unit Numbers [12/09/1998]
How do you prove this statement: For every prime number there exists a
repeated unit number that is a multiple of that prime.
- Primes and Squares [05/03/2001]
For what values of prime number p is (2^(p-1)-1)/p a perfect square?
- Primes Containing but Not Ending in 123456789 [02/26/2003]
Are there infinitely many primes that contain but do not end in the
block of digits 123456789 ?
- Primes Greater Than/Less Than Multiples of Six [01/18/2002]
Has the postulate stating that every prime number is either one more or
one less than a multiple of six, excluding 2 and 3, been proven?
- Primes in the Form n^2 + 1 [04/06/2003]
Let n be a positive integer with n not equal to 1. Prove that if n^2 + 1
is a prime, then n^2 + 1 is expressible in the form 4k + 1 with k in the
- Primes of the Form 4n+3 [11/07/1999]
Prove that there are infinitely many primes of the form 4n+3 where n is
an element of the natural numbers.
- Primes: p+1 a Multiple of 6? [10/06/2002]
Prove that if p and p+2 are both prime, then p+1 is divisible by 6.
Completely stuck on what to do.
- Primes that are Sums of Primes [06/22/1999]
Is there an nth prime number, p, (other than 5, 17 and 41) that is equal
to the sum of the prime numbers up to n? For example, the 7th prime is
- Primes That Are the Sum of 2 Squares [09/17/1999]
How can I prove that every prime of the form 4m + 1 can be expressed as
a sum of two squares?
- Prime Triplet [12/07/2001]
The consecutive odd numbers 3,5,7 are all primes. Are there infinitely
many such 'prime triplets'?
- Primitive Elements vs. Generators [05/24/2002]
Prove that x is a primitive element modulo 97 where x is not congruent
to 0 if and only if x^32 and x^48 are not congruent to 1 (mod 97).
- Primitive Pythagorean Triples [02/23/1998]
Given a triple of numbers (a, b, c) so that a, b, and c have no common
factors and satisfy a^2+b^2 = c^2, make a guess about when a, b, or c is
a multiple of 5.
- Primorials [10/15/2003]
We know that p_1 * p_2 * ... * p_n + 1 is either prime or divisible by
a prime not included in the list. But is the second condition
necessary? Is the result ever not prime?
- Probability of Divisibility [06/18/2002]
What is the probability that a randomly selected three-digit number
is divisible by 5?
- Probability of Random Numbers Being Coprime [08/12/1997]
I have heard that the probability of two randomly selected integers being
coprime is 6/(pi^2). How do you show this is true?
- Problem Posed by Fermat [05/04/2001]
Find a right triangle such that the hypotenuse is a square and the sum of
the two perpendiculars, or indeed of all three sides, is also a square...
- Product Always an Even Number? [03/17/2002]
The letters a1, a2, a3, a4, a5, a6, a7 represent seven positive whole
numbers; b1, b2, b3, b4, b5, b6, b7 represent the same numbers but in a
different order. Will the value of the product (a1-b1)(a2-b2)(a3- b3)(a4-
b4)(a5-b5)(a6-b6)(a7-b7) always be an even number?
- Product and Sum of Digits = Number [10/24/2001]
How many two-digit numbers exist such that when the products of their
digits are added to the sums of their digits, the result is equal to the
original two-digit number?
- Product of Primes [02/27/2002]
Can you provide me with the proof that every non-zero positive integer
can be written as a product of primes?
- Product of Two Primes [10/27/1999]
How many positive integers less than 100 can be written as the product of
the first power of two different primes?
- Products of Integers (Even or Odd) [03/23/2002]
How can I prove that the product of two even integers is an even integer
and the product of two odd integers is an odd integer?
- Programs to Find Prime Numbers [11/27/1996]
Can a program be written in BASIC to compute the number of prime numbers
smaller than n?
- Program to Convert Number Bases [07/12/1999]
Is there an easier method for converting bases than dividing and
collecting the remainders? I want to write a computer program to do this.
- Proof by Contraposition [03/06/2002]
How can I prove that n^6 + 2n^5 - n^2 - 2n is divisible by 120?
- Proof by Induction [05/24/2002]
Prove by induction that (n^7 - n) is divisible by 42.
- Proof by Mathematical Induction [09/24/1999]
Prove the following statement by mathematical induction: for any integer
n greater than or equal to 1, x^n - y^n is divisible by x-y where x and y
are any integers with x not equal to y.
- Proof Involving Legendre Symbol [02/03/2003]
If p, q are both prime odd numbers such that they are not factors of
a, and p=q(mod 4a), prove that (a/p)=(a/q).
- Proof Involving mod 5 [10/27/2002]
Prove n^2 mod 5 = 1 or 4 when n is an integer not divisible by 5.
- Proof of Lagrange's Theorem [11/23/2000]
I am looking for a proof of Lagrange's Theorem, which states that any
positive integer can be expressed as the sum of 4 square numbers.
- Proof of the Infinite Series That Calculates 'e' [02/04/2004]
Is there a proof about this infinite series that gives the value of e:
1 + 1/1! + 1/2! + 1/3! + 1/4! + . . . + 1/n! where n goes to infinity?
- Proof of the Rational Root Theorem [11/13/2000]
How can I prove the Rational Root theorem?
- Proof Regarding LCM [12/05/2001]
Is there a proof of the equation: given integers a and b, a*b = GCF(a,b)
- Proof That 0/0 = 1 Based on x^0 Equaling 1? [02/02/2006]
I know the reason x^0 = 1 is because 1 = (x^3)/(x^3) = x^(3-3) = x^0.
I also know that 0/0 doesn't make sense, but following a similar
argument for 0^0 gives 1 = (0^3)/(0^3) = 0^(3-3) = 0^0. But since
(0^3) = 0, haven't I just shown that 0/0 = 1?
- Proof That 2 Does Equal 1! [03/24/1997]
I came up with a proof that 1 = 2. Where does my math go wrong?
- Proof that an Even Number Squared is Even [06/02/1999]
How do you prove that any even number squared is even and any odd number
squared is odd?
- Proof That Equation Has No Integer Roots [05/09/2000]
How can I prove that if p is a prime number, then the equation x^5 - px^4
+ (p^2-p)x^3 + px^2 - (p^3+p^2)x - p^2 = 0 has no integer roots?
- Proof That Product is Irrational [03/28/2001]
How can I prove that the product of a non-zero rational number and an
irrational number is irrational without using specific examples?
- Proof That sin(5) is Irrational [04/24/2001]
How do you prove that sin(5) is an irrational number?