Solution Manual Introduction Number Theory Niven

Tenenbaum, Gérald Introduction to analytic and probabilistic number theory. Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, 1995.

Isn't it weird that nobody's mentioned so far Chandrasekharan's treatise on arithmetic functions? OK, they've already mentioned his book on the principles of analytic number theory, yet the book I'm now referring to is

Solution Manual Introduction Number Theory Niven

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The final two chapters of the text cover more advanced topics that are outside the mainstream of a typical undergraduate course and, I would guess, would almost never be reached in a course based on this book (but of course one cannot fault the author for including them and therefore making the text a bit more versatile): analytic techniques (including a discussion of the Riemann zeta function, the zeta function of an algebraic number field and the analytic class number formula), and the number field sieve, a relatively new (about 25 years old) factorization algorithm (the fastest one known, according to the author) which uses algebraic numbers. This chapter also contains, by way of introduction and explanation of why anybody would care about factorization, a brief discussion of the RSA cryptosystem. (For much more on factorization, see The Joy of Factoring by Wagstaff.)

I. Niven, H. S. Zuckerman, H. L. Montgomery, An introduction to the theory of numbers. W. J. LeVeque, Fundamentals of number theory. (With more historical remarks.) K. Ireland, M. Rosen, A classical introduction to modern number theory. (If you want to learn modern number theory, you can start with this book. Part of it will be studied in the Junior seminar guided by Professor Skinner. )

This is a tentative schedule for the course. If necessary, I might change it. 11 Sep: Introduction: Primes and factorization. Read: 1.2,1.3 15 Sep -- 19 Sep:Unique factorization, Euclid's algorithm, and Congruences.Read: 1.2 and 1.3 again22 Sep -- 26 Sep:Units and primes in a "System of numbers", and Euler's ϕ function: Euler's theorem and Fermat's ``little'' theorem. and modular arithmetic.Mod m numbers. Read: 2.129 Sep -- 3 Oct: Chinese Remainder Theorem, Number theory and public-key cryptography: the RSA algorithm.Read: 2.3, 2.5 6 Oct -- 10 Oct: Solutions of polynomials modulo prime powers, Hensel's lemma.Read: 2.6, 2.713 Oct -- 17 Oct:Multiplicative order, Primitive roots. Read: 2.87. 20 Oct -- 24 Oct:p-valuation, Primitive roots (continued), Quadratic residue, Midterm.Read: 2.8, 3.1 3 Nov -- 7 Nov:Polynomials over Z/pZ (review), Quadratic reciprocity.Read: 2.7, 3.29. 10 Nov -- 14 Nov: Quadratic reciprocity (continued), Great integer function, Arithmetic functions. Read: 3.2, 4.1, 4.217 Nov - 21 Nov:Convolution, and Möbius inversion.Read: 4.325 Nov:Geometry of numbers, Sum of two squares.Read: 6.41 Dec -- 5 Dec:Cyclotomic polynomials and its relation with primitive elements mod p, Beatty sequence and its generalization, Continued fractions.Read: 7.1, 7.2 8 Dec -- 12 Dec:Continued fractions (continued).Read: 7.3, 7.4, 7.5, 7.6

Due Sep. 25: Section 1.2, problems 47, 49, 51, 54; Section 1.3, problems 23, 36, 39; Due Oct. 2: Section 2.1, problems 10, 14, 20, 28, 44.Find all the Pythagorean triples, i.e. all triples of natural numbers (a,b,c) such that a2+b2=c2. (Hint: Use lines passing through (0,1) with rational slope. Rational points on the circle x2+y2=1. Use extra parameters, and give formulas for a, b, and c in terms of those extra parameters, e.g. one can write all the integer solutions of 3a+2b=1 as a=2t+1 and b=-3t-1, where t can be any integer number (convince yourself why?)) Describe all the primes in Z/mZ. In particular, say under what condition, there is only one prime, up to multiplication by a unit, in Z/mZ. Which prime numbers can be written as sum of two perfect squares? (Hint:Look at a2+b2 modulo 4. Use Wilson's theorem. As most of you have already noticed the hard part of this problem is in Lemma 2.13 of your book. I expect you to think on this problem, yourself. After a while, you can read proof of this lemma, understand it, and then answer this problem. Again let me emphasis that you have to write down your solution without looking at your book, i.e. you have to recreat the argument. ) Let n be an integer with g.c.d.(n,10)=1. Show that the decimal expansion of 1/n is repeating. (Hint: Look at the following equalities. 1/9 = 1/10 + 1/100 + 1/1000 + . . . = .11111 . . . 1/99 = 1/100 + 1/10000 + 1/1000000 + . . . = .01010101 . . . 1/999 = .001001001 . . . 1/9999 = .000100010001 . . . Use their generalization, coupled with another theorem, to get the desired result.) Due Oct. 9: Section 2.1, problems 29, 30, 36, 49. Section 2.3, problems 3, 7, 9, 13, 18, 21, 37, 41, 47. Due Oct. 16: Section 2.5: Problems 3, 4, 5. Section 2.6: Problems 2, 7. Section 2.7, Problem 10. Show that there is no positive integer n for which 2n + 1 is divisible by 7. Determine all integers n>1 such that (2n+1)/n2 is an integer. Let n>3 be an odd number. Show that there is p a prime number such that p 2&#981(n)-1, p does not divide n. (Hint: Take a look at: Probem 35, section 2.3, Problem 49, section 1.2, and Problem 43, section 1.3.) Due Nov. 6: Problems 7, 8, 9, 10 from the midterm. Section 2.8, Problems 20, 24, 29, 32. Due Nov. 14: Section 2.7, Problem 12. Section 2.8, Problems 33, 34, 35 Section 3.1, Problems 11, 15, 21, 23, 24 (There is a misprint. It should be (a,m)=1 instead of (a,p)=1.) Show that sum of quadratic residues mod p is divisible by p if p is an odd prime larger than 3. Due Dec. 2: (Happy Thanksgiving) (Extended to Dec. 4:) Section 3.1, Problem 20. Section 3.2, Problems 1, 7, 13, 15, 19, 20, 21, 24. Section 4.1, Problems 9, 20, 22, 23, 37. Section 4.2, Problems 5, 10, 16, 19, 25. Section 4.3, Problems 15, 23, 25, 26, 28, 29.Let &Phin(x) be the nth cyclotomic polynomial. a) Show that if p divides &Phin(x) for some integer x and (p,n)=1, then p is of the form nk+1. b) Use the first part to show that there are infinitely many primes of the form nk+1. Due Jan. 8: (Happy new year) a) Let p be prime larger than 3. For some integer x, p divides x2+x+1 if and only if p is of the form 3k+1. b) Show that x2+xy+y2=1 is an ellipse. Its axis are lines y=x and y=-x, and find its area. (Hint: Rotate by angle 45 degree. So x=(X+Y)/Sqrt(2) and y=(Y-X)/Sqrt(2), and rewrite the equation.) c) If p is a prime of the form 3k+1, then there are integers x and y such that p=x2+xy+y2. (Hint: Use parts a, b, and Minkowski's convex body.) Section 7.1, problem 3. Section 7.3, problem 3. Section 7.4, problem 1. Section 7.5, problems 3,4. Section 7.6, problems 3,4,5.

I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about the number theory and analysis, and I have been pursuing a mathematics major starting on this Spring semester. I have been independently self-studying the number theory textbooks written by Niven/Zuckerman/Montgomery, Apostol, and Ireland/Rosen on this semester. As this semester progressed, I discovered that I am more interested in the pure mathematics than applied aspects (computational biology, cryptography, etc.). I want to pursue a career as analytic number theorist and prove the Collatz conjecture and Erdos-Straus conjecture.

I have been thinking about doing the number-theory research on my university (research university; huge mathematics department). I have been self-studying the NT by myself and also regularly attending the professional and graduate seminars on number theory but I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them about if I can do undergraduate research under them? If research is not possible (perhaps due to my lacking maturity), should I request of doing independent reading under them and later proceed with the research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask to them or do in my own?

While it is possible for a highly competent mathematician to dole out a doable problem for an undergraduate to solve over the summer, empirical data has proven otherwise - i.e., it's rather hard for an undergrad to prove anything original in number theory if only given a few weeks during the summer.

Number theory is known to be a very difficult topic to get into. Before you decide to commit, take courses in complex analysis and abstract algebra. They are crucial if you want to read more advanced texts in number theory.

Course description:The first few weeks will be spent quickly covering the foundations of elementary number theory: divisibility, congruences, prime numbers, and so on, some of which might already be familiar to you. Once we have this foundation, many different subjects will be open to us, such as: finding roots of polynomial congruences; quadratic reciprocity; multiplicative functions. Topics that might also be covered include: running times of number-theoretic algorithms; RSA cryptography; writing numbers as sums of squares; Farey fractions and continued fractions; binary quadratic forms. I will also indicate the connections between these topics and other advanced areas of number theory (algebraic number theory, analytic number theory, diophantine approximation, etc.). 2ff7e9595c