Number theory 1. What is primitive root theorem? What is a primitive root root mod m
The primitive root theorem(See theorem 5.19, page. 144) states below:
,
Let n be a positive integer. Then a primitive root modulo n exists iff n is equal to 1, 2, 4, or 2 where p is an odd prime number and m is a positive integer. A primitive root mod m is a number x which order equals
, i.e. = .
(See page 130 for examples.) 2. If a primitive root root mod n exists, how many of them are they?
By corollary 5.6, page 132, If a primitive root mod n exists, there are exactly 3. If r is a primitive root root mod n,
= ,
( ( )
ones.
, = , then
, and { , , , … , } is a reduced residue system mod n.
Those are propositions 5.3 and 5.4, page 131 and 132. 4. Give you m such that a primitive root mod m exists. If
, = , How do you determine if a is an
nth power residue mod m? (Theorem 5.21)
By theorem 5.21, page 149, a is an nth power residue mod m iff /
= 1 ,
. (Furthermore, if a is an nth power residue module m, the congruence ≡ has exactly d incongruent solutions modulo m.
where d = (n,
5.
o prove a Diophan tine equation
,, ,, =
has no integral solution, it s enough to show that
,, ≡ for some m has no integral solutions. (Page 160) …if a Diophantine equation has a solution, then the corresponding congruence obtained by considering the equation modulo any positive integer m also has a solution. By contraposition then, if a positive integer m can be found so that a Diophantine equation viewed as an congruence modulo m has no solutions, then the original Diophantine equation also has no solutions. (See the page below for examples.) 6. What is Pythagorean triples? Describe all of them. Do some sim ple exercises.
The definition 3 at page 161: A triple x, y, z of positive integers satisfying the Diophantine equation +
= is said to be a Pythagorean triple. All the primitive Pythagorean
triples (i.e. (x, y, z)=1) can be given precisely by the equations
= = 2 = + Where m,n are integers, m>n>0, (m,n)=1, and exactly one of m and n is even. (This is theorem 6.3. See page 163 for proof of this theorem.)
7. N is a sum of two squares if and only if what? (See Theorem 6.8)
By theorem 6.8, page 171, N is the sum of two squares iff every prime factor congruent to 3 module 4 occurs to an even power in the prime factor of n. 8. All positive integers are expressible as the sum of four squares. How abo ut sum of five positiv squares?
The theorem of “All positive integers are expressible as the sum of four squares ” is theorem 6.13 (Lagrange). About sum of five positive squares, first, notice 169 = 13
= 12 + 5 = 12 + 4 + 3 = 8 + 8 + 5 + 4 . For all numbers K > 169, we set A = 169 , and by Lagrange theorem, there exists w, x, y, z, integers, such that A = + + + . Without loss of generosity, we may assume ≤ ≤ ≤ . We have four cases: > 0, = 13 + + + + . = 0 > 0, = 12 + 5 + + + . , = 0 > 0, = 12 + 4 + 3 + + . , , = 0 > 0, = 8 + 8 + 5 + 4 + . Thus every positive integers K >169 can be expressed as five positive squares. For positive integers K<169, we can check and find that only 1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, and 33 do not share this property. Reference: Extensions of a Sums-of-Squares Problem http://mathdl.maa.org/images/cms_upload/Extensions-Jackson_Masat_and_Mitchell34759.pdf
9. Review Fermat s little theorem.
Let p be a prime number and let a ∈Z. If p doesn ’t divide a, then − 10 Algorithm for compute continued fraction of an irrational numb er
Practice some examples e.g.
≡ 1 .
= → [ ] = = → [] = = → [] = + √ , √ ,.
This algorithm comes from the first half of proposition 7.13, page 204. For some simple examples, See example 10 and 11. 11 Given the periodic continued fraction of solutions to = ± if it exists.
√ = [… … ], review the formulas to find fundame ntal
For
= 1 , use theorem 8.6. Look at the period length of this expansion first, say p. If p is even, the fundamental solutions of this equation are given by ( , )=( −, − ). If p is odd, the fundamental solutions of this equation are given by ( , )=( −, − ). For = 1 , see page 283 for the answers to question 18. Look at the period length of this expansion first, say p. If p is even, this equation has no solutions. If p is odd, the fundamental solutions of this equation are given by ( , )=( −−, −− ).
