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Browse High School Number Theory
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Diophantine equations.
    Infinite number of primes?
    Testing for primality.
    What is 'mod'?

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) * LCM(a,b)?

Proofs that Every Natural Number Factors into Only One Unique Product of Primes [03/15/2012]
How do we know that every natural number has one, unique factorization? After identifying this as the Fundamental Theorem of Arithmetic and mentioning a proof by Euclid, Doctor Jacques sets up a simpler proof that relies on contradiction.

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?

Proof that Sqrt(3) is Irrational [08/14/1997]
How does one prove that sqrt(3) is irrational? or others? Is there a general algorithm? How about just for primes?

Proof That the Cube Root of 3 is Irrational [05/22/2000]
How can I show that the cube root of 3 is irrational?

Proof with Pigeonhole Principle [09/20/2001]
Prove that among five points selected inside an equilateral triangle with sides of length 2, there always exists a pair at a distance not greater than 1.

Proof with Powers of 2 and a Product [10/22/2008]
Prove that starting with any power of 2 there is a number such that the product of those two numbers will contain only the digits 1 and 2. For example, 4*3 = 12, 8 * 14 = 112, 64*33 = 2112.

Properties of the Phi Function [01/19/1999]
What are some properties of the phi function? What about the phi function and prime numbers?

Prove 101 the Only Prime [12/14/2002]
In this sequence of integers, all in base 10: 101, 10101, 1010101, 101010101, 10101010101, .......,,,, etc., prove that 101 is the only prime in the sequence.

Prove a and b are Perfect Squares [12/28/2001]
Let a and b be positive integers such that (a,b) = 1 and ab is a perfect square. Prove that a and b are perfect squares.

Prove a = b = c [01/27/2002]
When a^2 + b^2 + c^2 = ab + bc + ca and abc does not equal 0, prove that a = b = c.

Prove that 7 + 17*sqrt(17) is Irrational [09/22/2002]
I need to prove that 7 + 17*sqrt(17) is irrational. I know to set it equal to a/b, and have already proven that any square root of a prime number is irrational.

Prove That an Expression is a Multiple of 10 [12/19/2002]
If a and b are positive integers, prove that (a^5)*(b) - (a)*(b^5) is a multiple of 10.

Prove that Log A is Irrational [06/14/1998]
Can you help me prove that a common log of a number (not powers of 10) is irrational?

Prove that n^3 + 2n is Divisible by 3 [04/15/2001]
Prove by mathematical induction that n^3 + 2n is divisible by 3 for all positive integers.

Prove Twin Primes Greater Than 3 Divisible by 12 [10/08/2002]
Prove that if p and q are twin primes, each greater than 3, then p+q is divisible by 12.

Prove x^2+y^2 Not Divisible by 4 [09/20/2001]
Prove that if x and y are odd, then x^2 + y^2 is even but not divisible by 4.

Proving a Number is Prime [10/13/2004]
How do you really prove that 2 or some other number is a prime number?

Proving a^x = a^y iff x = y [12/13/2000]
How can I prove that a^x = a^y iff y = x for all real numbers x and y?

Proving De Moivre's Theorem [12/03/1997]
Prove De Moivres theorem: - (cos(x)+isin(x))^n = cos(nx) + isin(nx) .

Proving Divisibility [04/30/2002]
Prove that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 15.

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