Number Theory – Errata | A kind of library

Number Theory – Errata

This is a short and rather incomplete list of errata for Melvyn B. Nathanson’s Elementary Methods in Number Theory, Springer, Graduate Texts in Mathematics, vol. 195, (2000), MR1732941 (2001j:11001), compiled while teaching Math 507 – Advanced Number Theory in Fall 2010.

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 $M_m$ .

Update, January 4, 2018: The largest known prime is now $M_k$ for $k=77,232,917$ , a number with $23,249,425$ digits. It was identified on December 26, 2017. See here for the announcement and further details.

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. $B$ $\Longrightarrow$ $Bx$

2 Responses to Number Theory – Errata

Luca Goldoni says:

March 30, 2012 at 8:09 am

This is a great post. Thank you.

Luca Goldoni Ph.D University of Trento Italy

Reply
andrescaicedo says:

January 4, 2019 at 9:29 am

A new update: https://www.mersenne.org/primes/?press=M82589933 The current record for largest known prime is $M_k$ for $k=82,589,933$ .

Reply

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