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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'?
 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.
 Proving Divisibility [09/11/2003]

Prove that (n^2  n) is divisible by 2 for every integer n; that
(n^3  n) is divisible by 6; and that n^5  n is divisible by 30.
 Proving e is Irrational [11/19/1997]

My professor suggested using a proof by contradiction, but I don't
understand how to do it.
 Proving Fermat's Last Theorem for N = 4 [05/18/2000]

How can you prove Fermat's Last Theorem for the specific case n = 4?
 Proving O(n) [01/23/2001]

How would you prove that an equation is of order n, or n squared?
 Proving Perfect Squares [07/05/1998]

Suppose a, b, and c are positive integers, with no factor in common,
where 1/a + 1/b = 1/c. Prove that a+b, ac, and bc are all perfect
squares.
 Proving Phi(m) Is Even [04/22/1998]

Explain why phi(m) is always even for m greater than 2...
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