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MATH 100 - THE THEORY OF PRIMES AND FACTORIZATION
An element a in a number system is invertible if there exists b such that ab = 1.
Examples:
• In Z10 , the integers modulo 10, the invertibles are 1, 3, 7, 9.
• In the integers Z, the only invertibles are 1 and −1.
• In the Gaussian integers Z[i], the only invertibles are 1, −1, i and −i.
We don’t count invertible elements in factoring. There is a technical version of that
statement, omitted here.
A number is irreducible if it is not zero, not invertible and does not factor except
with one of the factors irreducible.
√ Irreducible is the same as prime in every system
we consider except one, viz., Z[ −3], so we use the terms interchangably.
• 2 and 5 are prime in Z10 . On the other hand, Z11 contains NO primes
because every element is invertible.
• 2, 3, 5, 7, 11, 13, 17, 19, . . . are primes in Z. We don’t count the negative
primes separately because −1 is invertible. Thus 7 and −7 are considered
equivalent for factoring purposes.
• 2 = (1 + i)(1 − i) is not prime in the Gaussian integers, but 3 is prime. More
on this later.
Factoring modulo n is interesting, but strange. For example, in Z10 we have
2 × 2 = 4 = 2 × 7, which are not equivalent factorizations!
Now we consider the ordinary integers Z. First note that every number can be
factored into prime numbers. In fact, 2k is the smallest number that factors into
k primes. So, for example, every number strictly less than 210 = 1024 factors into
somewhere between 1 and 9 prime factors.
Prime numbers are like atoms, the building blocks of numbers (multiplicatively).
Basic facts:
• There are infinitely many primes (Euclid).
• If a number is not prime, then one of its factors is at most its square root.
√
For example, 35 = 5 × 7 and 35 ≈ 5.9. So to show that
√ 137 is prime, we only
check that it is not a multiple of 2, 3, 5, 7 and ll because 137 ≈ 11.7.
Numbers can factor in more than one way. For example,
46, 189 = 143 × 323 = 209 × 221.
It is an amazing fact, due to Euclid, that when you factor all the way down to
primes, the factorization is unique. For example,
46, 189 = 11 × 13 × 17 × 19
and there is no other way to do it.
The fact that every number factors uniquely is known as the Fundamental Theorem of Arithmetic.
Date: February 5, 2008.
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MATH 100 - THE THEORY OF PRIMES AND FACTORIZATION
Now we turn to the Gaussian integers. Primes in Z[i] can be described as follows.
• 2 = (1 + i)(1 − i), and both those factors are prime.
• If an integer prime p satisfies p = 3 mod 4, then it is still prime in Z[i]. So
3, 7, 11, 17, 29, etc. are still prime.
• If an integer prime p satisfies p = 1 mod 4, then we can write p = a2 + b2 =
(a+bi)(a−bi) and both those factors are prime. For example, 13 = 32 +22 =
(3 + 2i)(3 − 2i) with 3 ± 2i prime.
Finally, we note that the Gaussian integers also have the unique factorization
property: they can be factored in Gaussian primes in essentially only one way.
√
The ring consisting of all complex numbers a + b −3 does not have unique
factorization. There we have
√
√
10 = 2 × 5 = (1 + −3)(1 − −3)
as two distinct factorizations of 10.
Resources
Sherman Stein, Mathematics: the man-made universe, Chapters 2 and 3.
Books on number theory, e.g., Elementary Number Theory by Charles Vanden
Enden.
Platonic Realms:
www.mathacademy.com/pr/prime/articles/fta/index.asp
Kevin Coombs, University of Texas:
odin.mdacc.tmc.edu/%7Ekrc/numbers/fta.html
Jim Loy:
www.jimloy.com/algebra/gprimes.htm
Harry J. Smith:
www.geocities.com/hjsimithh/GPrimes/GPriWhat.html