Ask Dr. Math

High School Archive

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'?

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Conjectures vs. Hypotheses [01/12/1999]

What is the difference between the terms 'conjecture' and 'hypothesis'? Should the Riemann hypothesis be the Riemann conjecture?

Connecting the Dots [03/14/1999]

If you have a few dots on a page, how many lines does it take to connect them all to each other?

Consecutive Composite Numbers [07/05/2002]

Find 1000 consecutive composite numbers.

Consecutive Composite Numbers [06/07/2005]

Is it possible to find m consecutive composite numbers where m is any natural number?

Consecutive Fibonacci Numbers Relatively Prime [11/17/2001]

Prove two consecutive Fibonacci numbers are relatively prime.

Consecutive Integers [02/10/1999]

Can the product of four consecutive integers be a perfect square?

Consecutive Integers, Divisible by the Sums of Their Digits [05/20/2002]

Find the first three consecutive integers greater than 10, each of which is divisible by the sum of its digits.

Consecutive Non-Prime Integers [2/17/1996]

For any integer n > 1 there exist n consecutive non prime integers. I have taken n = 4 and then used 24,25,26,27 as a specific example. However, for n large I don't see how this can hold.

Continued Fraction for Tan(x) [11/03/2000]

Can you tell me why the "infinite continued fraction representation" for the tangent function works?

Continued Fractions [10/07/1997]

Exactly what are "continued fractions"?

Convergence of a Prime Sequence [02/10/2002]

We know the sum(1/n) (n=1, infinity) does not converge, but what about sum(1/p) where sum is over prime numbers only. Does it converge?

Converting a Base 2 Log [01/31/2002]

2147483647 is one less than what power of 2?

Converting a Fraction Into a Continued Fraction [11/29/2003]

I know how to convert a continued fraction into a standard fraction, but I can't figure out how to to the other way. How can I convert a standard fraction into a continued one?

Converting any Number Base to Another [05/11/2001]

Is there a particular technique for the conversion of any number base into any other base? I also need information on basic numeric calculations within a base 5 system.

Converting Base Numbers Directly Without Using Base 10 [01/26/2004]

How do you convert from one base to another without going through base 10? How do I go from base 3 to base 7, for instance? Would the same algorithm work to go from base 5 to base 12?

Converting Bases [9/5/1996]

How do I convert from one base to another - i.e. decimal to bin, hex, and oct?

Converting Bases [12/21/1997]

How do you convert hexadecimal, binary, and decimal numbers?

Converting Bases Without Going through Base 10 [09/04/2002]

Is there a way to convert between bases without going through base 10?

Converting Binary and Decimal to Hexadecimal and Back [08/12/2001]

How do you convert binary and decimal numbers to hexadecimal and vice versa?

Converting Binary to and From Decimal [06/16/1999]

Can you show me how to convert base 2 numbers into base 10 and vice- versa?

Converting Decimal Numbers from Base 2 to Base 8 or 16 [02/20/2009]

I understand converting binary to octal or hexadecimal in general, but am curious how you handle the decimal portion of a number.

Converting Directly from Base to Base [04/01/2001]

Is there a way of reducing the representation of a natural number from one base to another directly without reducing to base 10 first?

Converting Floating Decimals to Other Bases [11/27/2003]

I know how to convert integers to other bases. But how do I convert the decimal parts of non-integers such as 123.5 or 8.4444....?

Converting Fractions from Binary to Decimal [02/25/2002]

Can you explain how to convert binary fractions to decimal numbers, e.g. 0.00011001100110011001...?

Converting from Base 10 to Base 3 [02/12/2002]

How do you convert a base 10 number (example 2315) to base 3?

Converting from One Base to Another [10/28/1996]

How do you convert a number base 9 to base 4? from base 9 to base 10?

Converting Negative Decimals to Hexadecimal [05/30/2001]

Using a calculator I can get conversions like - 16 = F0, but what I need to know is how to get to F0 from - .16

Converting Numbers: Binary to Decimal [11/20/1997]

I just cannot convert from binary to decimal and back again.

Converting Repeating Decimals from Base X to Base Y [10/02/2009]

How do I convert a repeating decimal from base 9 to base 6? For example, 0.54444444... (base 9) to base 6.

Converting to Base 16; Place Value Chart [03/22/1998]

How do you convert numbers to base 16 numbers?

Converting to Binary [05/27/2001]

Converting the number .78125 to its binary equivalent.

Coprimes in Fermat's Last Theorem [06/03/1999]

Why are (z-x)/2 and (z+x)/2 coprime in Fermat's Conjecture when n = 2?

Countability of Rational and Irrational Numbers [01/30/2001]

When speaking of countability of numbers, which has more, rational or irrational?

Counting Digits [03/13/2002]

How many times does each digit appear when counting from 1-1000 and 1- 10000 (including zeros)?

Counting Even Digits in Three-Digit Numbers [10/24/2004]

How many 3-digit numbers are there in which the number of even digits is even?

Counting in Base 5 [03/02/2004]

How do I count from 1 to 7 in base 5?

Counting Odd Coefficients [05/27/1998]

If (1+x)^100 is multiplied out, how many of the coefficients are odd? How would you generalize?

Counting Positive Rational Numbers [09/09/2001]

In Hardy's book _Pure Mathematics_ he gives a formula for counting the positive rational numbers p/q when they are arranged in a triangular matrix and counted down diagonally from the top row... how can it be proved for all such numbers?

C++ Program to Convert Decimal to Binary [10/26/1999]

Can you show me a C++ routine that converts decimal numbers to binary?

Cubes as Differences of Squares [07/04/2002]

Prove that the cube of any positive integer is equal to the difference of the squares of two integers.

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