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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 twodigit 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 twodigit number?
 Product of Primes [02/27/2002]

Can you provide me with the proof that every nonzero 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 xy 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^(33) = 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^(33) = 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^2p)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 nonzero 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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