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'?
- Pairs of Integers [08/16/1997]
Show that there are infinitely many pairs of integers(x,y) such that
x|y**2+m and y|x**2+m where m is any chosen integer; moreover gcd(x,y)=1.
- Paradox in the Zero Power [09/26/2000]
Why does the product of multiplying a number by itself zero times equal
- Partitioning an Integer [11/14/1998]
How many different ways are there of making a number by adding different
combinations of three numbers?
- Partitions and Products [01/02/2003]
What is the best way of dividing an integer into parts so that the
product of the parts will be as large as possible? Is there a
universal law that tells us what partition will produce the maximum
product for any given number? And can such a law be proved?
- Pascal's Triangle and Fibonacci Formula [02/23/2002]
Prove that the diagonals of Pascal's triangle are the Fibonacci numbers.
- Paths to Triangle Points [05/26/1999]
How can I find the number of paths to a point using Pascal's triangle?
- Pattern in Period [01/31/1999]
Is there any pattern in odd periods with 1 as numerator?
- Patterns in Repeating Decimals [08/06/2003]
Why do certain number sequences repeat in the decimal expansions of
- Patterns in Rolling 3 Dice [09/16/1997]
I have come out with 216 outcomes when rolling 3 dice; while I was
listing all the outcomes, I began to see a pattern...
- Pell Pairs of Positive Integer Solutions [03/29/2012]
Solving a Pell equation confuses a student. Doctor Vogler applies modular arithmetic to
show the way, then concludes with a survey of online resources into Diophantine
- Percentage of Numbers Divisible by 6, 8 [03/18/2003]
What percentage of numbers is divisible by 6? by 8?
- Percentages of Prime Numbers [01/31/1999]
Does the percentage of prime numbers at every power of 10 decrease until
it reaches a constant value?
- Perfect Logarithms [06/24/2002]
What can you tell me about the equation log(abc)= log(a+b+c)?
- Perfect Number [7/21/1996]
How can I find a perfect number? What are some reference books about
- Perfect Number Algorithms [03/31/1998]
What is the formula for a computer program that tests whether an integer
is a perfect integer?
- Perfect Numbers [11/10/1997]
Do the sums of the digits of perfect numbers always equal 1?
- Perfect Numbers [06/15/2002]
Please show that any even perfect number ends in 6 or 8.
- Perfect Square [10/26/2001]
If a and b are positive integers such that (1+ab) divides (a^2+b^2), show
that the integer (a^2+b^2)/(1+ab) must be a perfect square.
- Perfect Square [07/23/2001]
If g.c.d.(x, 3) = 1 and g.c.d.(y, 3) = 1, show that x^2 + y^2 cannot be
a perfect square.
- Perfect Square? [02/18/2002]
If we use the digits 1,2,3,4,5,6,7 each only once to form a 7-digit
number, can the resulting number be a perfect square?
- Perfect Square, Cube, Fourth Power [01/25/2002]
Find the least integer greater than 1 that is a perfect square, a perfect
cube, and a perfect fourth power.
- Perfect Square Equation [02/22/2002]
Prove that if n is greater than 1, then nC2 + (n-1)C2 is a perfect
- Perfect Squares and Irrational Numbers [02/13/2002]
Isn't any non-perfect square an irrational number? What is the number
0.49? Its square root is 0.7, which is neither irrational nor an integer.
- Perfect Squares: n+125 and n+201 [01/21/2002]
Find the smallest positive integer n so that n+125 and n+201 are both
- Perfect Squares with Congruences [02/16/2002]
Prove that there is no perfect square a^2 whose last digits are 35.
- Perimeter of Pascal's Triangle [06/05/2001]
Is there a general formula for finding the perimeter of Pascal's triangle
using the number of rows?
- The Phi Function [11/21/1998]
What are the conditions on n,m so that phi(n*m) = phi(n)*phi(m)? What is
- Polynomial Divisible by 7 [11/14/2001]
Prove that 2^(3n+1) + 4^(3n+1) + 1 is divisible by 7.
- A Polynomial in Three Variables with Few Integer Solutions [03/12/2011]
A student seeks proof that a polynomial in n and two other variables has no integer solutions. After a little insight from modular arithmetic and a lot of searching with a computer algebra system, Doctor Vogler turns up many solutions.
- Positives and Negatives with Infinity and Zero [10/05/1999]
Are there such things in math as +0, -0, and unsigned 0; and +infinity, -
infinity, and unsigned infinity? Are these different?
- Powers of 2 Proof [03/24/2003]
Prove that any number that is not a power of 2 can be expressed as a
sum of two or more consecutive positive integers, but that this is
not possible for powers of 2.
- p, p+8, p+22 Not Prime [10/16/2001]
Prove that there is no positive integer p such that each of the numbers:
p, p+8, p+22 is prime.
- Prefix for 10^30 Bytes [05/25/2000]
What do you call 1,000,000,000,000,000,000,000,000,000,000 or 10^30
- Primality Test [11/26/2001]
I want to write a program using Pascal that will verify whether a number
- Primality Testing [04/22/1998]
Is there any fomula to find if a number is a prime?
- Primality Testing [12/02/2004]
How can I determine if a given number is prime?
- Prime and Consecutive Numbers [11/16/2001]
Why are 3, 5, and 7 the only numbers that appear to be prime and
- Prime Factor [02/18/2002]
I need to prove that each integer of the form 3n + 2 has a prime
factor of this form.
- Prime Factors, Modular Arithmetic, and Using Pari [08/08/2007]
Suppose we have two positive integers 'a' and 'b'. Is there a method
to find a positive integer 'k', such that a + bk = x^2 for some
integer 'x'? In other words, how can we find a positive integer 'k',
such that 'a + bk' is a square?
- Prime Factors of 4,194,305 [09/20/1999]
How can I find the prime factors of 2^22+1?