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# Category Archives: modular arithmetic

## MaBloWriMo 9: omega and its ilk

So far, we have defined a sequence of numbers , and showed that where and . This is a big step: the are defined recursively (that is, each is defined in terms of the previous ), but and give us … Continue reading

Posted in algebra, arithmetic, modular arithmetic, number theory
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, prime

## MaBloWriMo 8: definition of s and mod

I was a little unsatisfied with my proof yesterday since I don’t think I did a very good job explaining how enters into things. When sinuheancelmo asked a question which seemed to show confusion on exactly that point, I figured … Continue reading

Posted in arithmetic, iteration, modular arithmetic, number theory
Tagged definition, lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test

## MaBloWriMo 7: s via omega

Yesterday, I challenged you to prove that where , , and the are defined by and . The proof is by induction on . The base case is just arithmetic: Now suppose that we already know the statement holds for … Continue reading

Posted in algebra, arithmetic, iteration, modular arithmetic, number theory
Tagged induction, lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
11 Comments

## MaBloWriMo 5: The Lucas-Lehmer Test

We now know that can only be prime when is prime; but even when is prime, sometimes is prime and sometimes it isn’t. The Lucas-Lehmer test is a way to tell us whether is prime, for any prime . The … Continue reading

Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
3 Comments

## MaBloWriMo 4: not all prime-index Mersenne numbers are prime

Over the past couple days we saw that if is composite, then is also composite. Equivalently, this means that if we want to be prime, then at the very least must also be prime. But at this point there is … Continue reading

Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment

## MaBloWriMo 3: Mersenne composites in binary

Yesterday we saw that must be composite, since . Today I’ll talk about a somewhat more intuitive way to see this. Recall that we can write numbers in base 2, or “binary”, using the digits 0 and 1 (called “bits”, … Continue reading

Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment

## MaBloWriMo: Mersenne composites

The name of the game is to find Mersenne numbers which are also prime. Today, a simple observation: can only be prime when is also prime. Put conversely, if is composite then is also composite. For example, is composite and … Continue reading

Posted in algebra, arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes
Tagged lehmer, lucas, MaBloWriMo, Mersenne, prime, proof, test
1 Comment