# Category Archives: Number Theory

## Q: How does “1+2+3+4+5+… = -1/12” make any sense?

Physicist: When wondering across the vast plains of the internet, you may have come across this bizarre fact, that , and immediately wondered: Why isn’t it infinity?  How can it be a fraction?  Wait… it’s negative? An unfortunate conclusion may … Continue reading

Posted in -- By the Physicist, Equations, Math, Number Theory | 14 Comments

## Q: When you write a fraction with a prime denominator in decimal form it repeats every p-1 digits. Why?

The original question was: How come the length of the repetend for some fractions (e.g. having a prime number p as a denominator) is equal to p-1? Physicist: The question is about the fact that if you type a fraction … Continue reading

Posted in -- By the Physicist, Math, Number Theory | 8 Comments

## Q: How good is the Enigma code system compared to today’s publicly available cryptography systems?

Physicist: Freaking terrible. The Enigma machine used a “rolling substitution cypher” which means that it was essentially a (much more) complicated version of “A=1, B=2, C=3, …”.  The problem with substitution cyphers is that if parts of several messages are … Continue reading

## Q: Is there a formula for finding primes? Do primes follow a pattern?

Physicist: Primes are, for many purposes, basically random.  It’s not easy to “find the next prime” or determine if a given number is prime, but there are tricks.  Which trick depends on the size of the number.  Some of the … Continue reading

Posted in -- By the Physicist, Math, Number Theory | 31 Comments

## Learning intro number theory

Physicist: We occasionally get questions about free learning resources.  Khan academy is excellent, and if you poke around you can find a smattering of free class notes and text books, but generally speaking the more more detailed/advanced the material, the … Continue reading

Posted in -- By the Physicist, Brain Teaser, Math, Number Theory | 5 Comments

## Q: How do you write algorithms to enycrypt things?

Physicist: There are several algorithms, but almost all of them are all based on “trap-door encryption”.  The idea is that you find some kind of mathematical process that’s easy to run forward, but effectively impossible to run backward, unless you … Continue reading