Number Theory – Errata

This is a list in progress of Errata for Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, Graduate Texts in Mathematics, vol. 195, (2000). I will be adding to the list as I find time; please let me know of additional typos/corrections you may find.

Chapter 1

Page 7. Exercise 8. that n \Longrightarrow that if n

Page 8. Exercise 9. that n \Longrightarrow that if n

Page 9. Exercise 25.(a). reflexive in \Longrightarrow reflexive (or anti-symmetric) in

Page 10. Line break needed in line 4.

Page 12. Line -11. every nonempty \Longrightarrow every such nonempty

Page 12. Line -8. integers has \Longrightarrow integers, not all 0, has

Page 14. Line -9. integers \Longrightarrow integers, not all 0

Page 20. Line break needed in line -11.

Page 31. Line 11. mp_i^{-k_i} \Longrightarrow mp_i^{-r_i}

Page 36. Line 1. integers \Longrightarrow integers with n>1

Page 37. Line 12. 14 \Longrightarrow 13

Page 40. Line 12. x_1 \Longrightarrow x_2

Page 41. Line break needed in line 8.

Page 43. The last paragraph can be updated as follows: By September, 2010, a few more Mersenne primes have been found. The list continues with M_n=2^n-1 for n=13,466,917; 20,996,011; 24,036,583; 25,964,951; 30,402,457; 32,582,657; 37,156,667; 42,643,801; and m=43,112,609. It is not known whether this list includes all Mersenne primes less than or equal to M_m, or if some have been skipped. The largest known prime is M_m.

Chapter 2

Page 48. Line 6. m{\mathbf Z}) \Longrightarrow m{\mathbf Z}

Page 53. Line 14. to \Longrightarrow for

Page 57. Line 17. \varphi(2)=2 \Longrightarrow \varphi(2)=1

Page 57. Line 18. \varphi(3)=3 \Longrightarrow \varphi(3)=2

Page 58. Line breaks needed in lines -7 and -4.

Page 61. Exercise 11. \varphi(p^k)=\varphi(p) \Longrightarrow f(p^k)=f(p)

Page 68. Line -15. p \Longrightarrow m

Page 72. Exercise 6. 7 into 1 \Longrightarrow 1 into 7

Page 78. Line 7. m If \Longrightarrow m. If

Chapter 3

Page 90. Line break needed in line 1.

Page 91. Line 5. F[x] \Longrightarrow F[x], not both 0

Page 95. Line 10. of theorem \Longrightarrow of the theorem

Page 98. Line 5. \displaystyle \frac{r_n}{3^n} \Longrightarrow \displaystyle \frac{r_n}{2^n}

Page 98. Line 14. 5, 4 \Longrightarrow 5

Page 99. Line -4. \pmod3 \Longrightarrow \pmod {19}

Page 108. Line -2. \left(\frac{q}{u_1!\dots u_k!}\right) \Longrightarrow \binom{q}{u_1,\dots,u_k}=\frac{q!}{u_1!\dots u_k!}

Page 108. Line -1. \displaystyle \left(\frac{q}{u_1!\dots u_k!}\right) \Longrightarrow \displaystyle \binom{q}{u_1,\dots,u_k}

Page 109. Line break needed in line 13.

Page 114. Exercise 5. quaratic \Longrightarrow quadratic

Page 114. Exercise 6. quaratic \Longrightarrow quadratic

Page 118. Line -13. t \Longrightarrow x

Chapter 4

Page 122. Line 8. G(p) \Longrightarrow G(p_i)

Page 124. Line 2. \oplus\cdots\oplus \Longrightarrow +\cdots +

Page 124. Line 12. G_1\oplus+\cdots+\oplus G_k \Longrightarrow G_1\oplus\cdots\oplus G_k

Page 125. Exercise 8(b). \cdots r_k \Longrightarrow \cdots=r_k

Page 125. Line break needed in line -2.

Page 126. Line -6. =\chi(g)\chi^{-1}(g) \Longrightarrow \chi(g)\chi^{-1}(g)

Page 127. Line 2. \chi(g)\bar\chi)(g) \Longrightarrow \chi(g)\bar\chi(g)

Page 128. Line -2. \chi_i(g_1)=1 for all g_i\in G_i \Longrightarrow \chi_i(h)=1 for all h\in G_i

Page 134. Line 11. \displaystyle \sum x\in G f(x) \Longrightarrow \displaystyle \sum_{x\in G}f(x)

Page 134. Line 12. L^2(G) defined \Longrightarrow L^2(G) is defined

Page 137. Line -1. \displaystyle \sum_{\chi\in\text{\normalsize supp}(\hat f)} \Longrightarrow \displaystyle \sum_{\chi\in\text{supp}(\hat f)}

Page 138. Line 2. \displaystyle \sum_{\chi\in\text{\normalsize supp}(\hat f)} \Longrightarrow \displaystyle \sum_{\chi\in\text{supp}(\hat f)}

Page 138. Line 5. (Twice) \displaystyle \sum_{\chi\in\text{\normalsize supp}(\hat f)} \Longrightarrow \displaystyle \sum_{\chi\in\text{supp}(\hat f)}

Page 138. Line 5. = \Longrightarrow \le

Page 138. Lines 7, 8. \displaystyle \sum_{\chi\in\text{\normalsize supp}(\hat f)} \Longrightarrow \displaystyle \sum_{\chi\in\text{supp}(\hat f)}

Page 138. Line -8. supp \Longrightarrow \text{supp}

Page 140. Line 6. \underbrace{\chi*\cdots*\chi}_{\displaystyle k \text{\normalsize \ times}} \Longrightarrow \underbrace{\chi*\cdots*\chi}_{k \text{\ times}}

Page 140. Line 6. x_k) \Longrightarrow x_k)=\chi(a)|G|^{k-1}

Page 140. Line 10. \underbrace{\ell_p*\cdots*\ell_p}_{\displaystyle k \text{\normalsize \ times}} \Longrightarrow \underbrace{\ell_p*\cdots*\ell_p}_{k \text{\ times}}

Page 142. Line 9. f^\sharp is \Longrightarrow f^\sharp evaluated at \chi^\sharp is

Page 142. Line -6. It \Longrightarrow Since \pi^\sharp is an isomorphism, it

Page 143. Exercise 3. G/H \Longrightarrow G/H

Page 145. Line 5. {\mathcal B} \Longrightarrow {\mathcal B}'

Page 146. Line -15. {\mathcal B} \Longrightarrow {\mathcal B}'

Page 147. Theorem 4.14. group \Longrightarrow group of order n

Page 149. Theorem 4.15. For \Longrightarrow Let G be a finite abelian group of order n. For

Page 149. Line -2. then \Longrightarrow then (with n=|G|)

Page 150. Line -7. C_{y^\sharp} \Longrightarrow C_{f^\sharp}

Page 151. Line 2. \bar\chi(x),= \Longrightarrow \bar\chi(x)=

Page 154. Theorem 4.18. is prime \Longrightarrow is an odd prime

Page 155. Lines 3 (twice), 4, 5, 6, 8, 11. -a \Longrightarrow a

Page 157. Line 9. 10 \Longrightarrow 13

Page 157. Line -8. \underbrace{\hat{\ell_p}*\cdots*\hat{\ell_p}}_{\displaystyle k \text{\normalsize \ times}}=\widehat{\underbrace{\ell_p*\cdots*\ell_p}_{\displaystyle k \text{\normalsize \ times}}} \Longrightarrow \underbrace{\hat{\ell_p}\times\cdots\times\hat{\ell_p}}_{k \text{\ times}}=\widehat{\underbrace{\ell_p*\cdots*\ell_p}_{k \text{\ times}}}

Page 157. Line -6. \widehat{\widehat{\underbrace{\ell_p*\cdots*\ell_p}_{\displaystyle k \text{\normalsize \ times}}}} \Longrightarrow \widehat{\widehat{\underbrace{\ell_p*\cdots*\ell_p}_{k \text{\ times}}}}

Page 157. Lines -5, -3. \underbrace{\ell_p*\cdots*\ell_p}_{\displaystyle k \text{\normalsize\ times}} \Longrightarrow \underbrace{\ell_p*\cdots*\ell_p}_{k \text{\ times}}

Page 160. Line -10. a a \Longrightarrow a

Page 162. Line break needed in line -5.

Page 163. Line break needed in line 2.

Page 163. Line 12. \frac{(k-j)\pi}{n}) \Longrightarrow \frac{(k-j)\pi}{n}

Page 169. Line 5. appears \Longrightarrow appears in

Chapter 5

Page 174. Line -8. \displaystyle\bigcap_{I\in{\displaystyle\text{Spec}}(R)} \Longrightarrow \displaystyle \bigcap_{I\in\text{Spec}(R)}

Chapter 6

Page 220. Remove line 4.

Chapter 8

Page 271. Line 3. B \Longrightarrow Bx

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One Response to Number Theory – Errata

  1. Luca Goldoni says:

    This is a great post. Thank you.

    Luca Goldoni Ph.D University of Trento Italy

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