Reference request: Oldest number theory books with (unsolved) exercises?
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Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and Hardy.
Motivation for this question. Some person came up to me before class and asked, are you the person asking the ridiculous "oldest book with exercises" questions on MO? I said yes, and he asked I could do one on number theory. So here we are.
nt.number-theory reference-request
$endgroup$
add a comment |
$begingroup$
Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and Hardy.
Motivation for this question. Some person came up to me before class and asked, are you the person asking the ridiculous "oldest book with exercises" questions on MO? I said yes, and he asked I could do one on number theory. So here we are.
nt.number-theory reference-request
$endgroup$
2
$begingroup$
Elementary Number Theory by Uspensky and Heaslet has a bunch of problems in each chapter. Not sure whether it's the oldness you are looking for (my impression is that the exercises don't get better if you go older than this).
$endgroup$
– darij grinberg
yesterday
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
yesterday
2
$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
yesterday
$begingroup$
Do you mean "unsolved" within the book itself, or "unsolved" anywhere?
$endgroup$
– James
17 hours ago
add a comment |
$begingroup$
Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and Hardy.
Motivation for this question. Some person came up to me before class and asked, are you the person asking the ridiculous "oldest book with exercises" questions on MO? I said yes, and he asked I could do one on number theory. So here we are.
nt.number-theory reference-request
$endgroup$
Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and Hardy.
Motivation for this question. Some person came up to me before class and asked, are you the person asking the ridiculous "oldest book with exercises" questions on MO? I said yes, and he asked I could do one on number theory. So here we are.
nt.number-theory reference-request
nt.number-theory reference-request
edited yesterday
community wiki
Get Off The Internet
2
$begingroup$
Elementary Number Theory by Uspensky and Heaslet has a bunch of problems in each chapter. Not sure whether it's the oldness you are looking for (my impression is that the exercises don't get better if you go older than this).
$endgroup$
– darij grinberg
yesterday
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
yesterday
2
$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
yesterday
$begingroup$
Do you mean "unsolved" within the book itself, or "unsolved" anywhere?
$endgroup$
– James
17 hours ago
add a comment |
2
$begingroup$
Elementary Number Theory by Uspensky and Heaslet has a bunch of problems in each chapter. Not sure whether it's the oldness you are looking for (my impression is that the exercises don't get better if you go older than this).
$endgroup$
– darij grinberg
yesterday
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
yesterday
2
$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
yesterday
$begingroup$
Do you mean "unsolved" within the book itself, or "unsolved" anywhere?
$endgroup$
– James
17 hours ago
2
2
$begingroup$
Elementary Number Theory by Uspensky and Heaslet has a bunch of problems in each chapter. Not sure whether it's the oldness you are looking for (my impression is that the exercises don't get better if you go older than this).
$endgroup$
– darij grinberg
yesterday
$begingroup$
Elementary Number Theory by Uspensky and Heaslet has a bunch of problems in each chapter. Not sure whether it's the oldness you are looking for (my impression is that the exercises don't get better if you go older than this).
$endgroup$
– darij grinberg
yesterday
1
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
yesterday
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
yesterday
2
2
$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
yesterday
$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
yesterday
$begingroup$
Do you mean "unsolved" within the book itself, or "unsolved" anywhere?
$endgroup$
– James
17 hours ago
$begingroup$
Do you mean "unsolved" within the book itself, or "unsolved" anywhere?
$endgroup$
– James
17 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
I wonder if you are already aware of R. D. Carmichael's "The theory of numbers" (John Wiley & Sons, Inc., NY, pp. 94, 1914.).
Apropos of the exercises in this monograph, one can read the following in the preface:
Numerous problems are supplied throughout the text. These have been
selected with great care so as to serve as excellent exercises for the
student's introductory training in the methods of number theory and to
afford at the same time a further collection of useful results. The
exercises with a star are more difficult than the others; they will
doubtless appeal to the best students.
Among the numerous problems supplied, the eighth problem on page 36 does stand out because, as far as I know, nobody has been able to solve it yet. It goes as follows:
- Show that if the equation $$phi(x) = n$$ has one solution; it always has a second solution, $n$ being given and $x$ being the
unknown.
Oddly enough, Carmichael didn't consider that this question deserved a star... In case you want to learn more about the history of this problem, I recommend that you take a look at the following installment of The evidence (a column that Stan Wagon used to contribute to The Mathematical Intelligencer):
S. Wagon, Carmichael's "empirical theorem". Math. Intelligencer, 8 (1986), No. 2, pp. 61-63.
$endgroup$
1
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
yesterday
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
yesterday
2
$begingroup$
@user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
23 hours ago
$begingroup$
Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory.
$endgroup$
– Gerry Myerson
23 hours ago
add a comment |
$begingroup$
The book "Théorie des nombres, Tome premier" by Edouard Lucas was published 1891. Many of the "Exemples" are actually exercises left to the reader. A scan is freely available in the archive.
$endgroup$
add a comment |
Your Answer
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$begingroup$
I wonder if you are already aware of R. D. Carmichael's "The theory of numbers" (John Wiley & Sons, Inc., NY, pp. 94, 1914.).
Apropos of the exercises in this monograph, one can read the following in the preface:
Numerous problems are supplied throughout the text. These have been
selected with great care so as to serve as excellent exercises for the
student's introductory training in the methods of number theory and to
afford at the same time a further collection of useful results. The
exercises with a star are more difficult than the others; they will
doubtless appeal to the best students.
Among the numerous problems supplied, the eighth problem on page 36 does stand out because, as far as I know, nobody has been able to solve it yet. It goes as follows:
- Show that if the equation $$phi(x) = n$$ has one solution; it always has a second solution, $n$ being given and $x$ being the
unknown.
Oddly enough, Carmichael didn't consider that this question deserved a star... In case you want to learn more about the history of this problem, I recommend that you take a look at the following installment of The evidence (a column that Stan Wagon used to contribute to The Mathematical Intelligencer):
S. Wagon, Carmichael's "empirical theorem". Math. Intelligencer, 8 (1986), No. 2, pp. 61-63.
$endgroup$
1
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
yesterday
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
yesterday
2
$begingroup$
@user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
23 hours ago
$begingroup$
Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory.
$endgroup$
– Gerry Myerson
23 hours ago
add a comment |
$begingroup$
I wonder if you are already aware of R. D. Carmichael's "The theory of numbers" (John Wiley & Sons, Inc., NY, pp. 94, 1914.).
Apropos of the exercises in this monograph, one can read the following in the preface:
Numerous problems are supplied throughout the text. These have been
selected with great care so as to serve as excellent exercises for the
student's introductory training in the methods of number theory and to
afford at the same time a further collection of useful results. The
exercises with a star are more difficult than the others; they will
doubtless appeal to the best students.
Among the numerous problems supplied, the eighth problem on page 36 does stand out because, as far as I know, nobody has been able to solve it yet. It goes as follows:
- Show that if the equation $$phi(x) = n$$ has one solution; it always has a second solution, $n$ being given and $x$ being the
unknown.
Oddly enough, Carmichael didn't consider that this question deserved a star... In case you want to learn more about the history of this problem, I recommend that you take a look at the following installment of The evidence (a column that Stan Wagon used to contribute to The Mathematical Intelligencer):
S. Wagon, Carmichael's "empirical theorem". Math. Intelligencer, 8 (1986), No. 2, pp. 61-63.
$endgroup$
1
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
yesterday
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
yesterday
2
$begingroup$
@user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
23 hours ago
$begingroup$
Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory.
$endgroup$
– Gerry Myerson
23 hours ago
add a comment |
$begingroup$
I wonder if you are already aware of R. D. Carmichael's "The theory of numbers" (John Wiley & Sons, Inc., NY, pp. 94, 1914.).
Apropos of the exercises in this monograph, one can read the following in the preface:
Numerous problems are supplied throughout the text. These have been
selected with great care so as to serve as excellent exercises for the
student's introductory training in the methods of number theory and to
afford at the same time a further collection of useful results. The
exercises with a star are more difficult than the others; they will
doubtless appeal to the best students.
Among the numerous problems supplied, the eighth problem on page 36 does stand out because, as far as I know, nobody has been able to solve it yet. It goes as follows:
- Show that if the equation $$phi(x) = n$$ has one solution; it always has a second solution, $n$ being given and $x$ being the
unknown.
Oddly enough, Carmichael didn't consider that this question deserved a star... In case you want to learn more about the history of this problem, I recommend that you take a look at the following installment of The evidence (a column that Stan Wagon used to contribute to The Mathematical Intelligencer):
S. Wagon, Carmichael's "empirical theorem". Math. Intelligencer, 8 (1986), No. 2, pp. 61-63.
$endgroup$
I wonder if you are already aware of R. D. Carmichael's "The theory of numbers" (John Wiley & Sons, Inc., NY, pp. 94, 1914.).
Apropos of the exercises in this monograph, one can read the following in the preface:
Numerous problems are supplied throughout the text. These have been
selected with great care so as to serve as excellent exercises for the
student's introductory training in the methods of number theory and to
afford at the same time a further collection of useful results. The
exercises with a star are more difficult than the others; they will
doubtless appeal to the best students.
Among the numerous problems supplied, the eighth problem on page 36 does stand out because, as far as I know, nobody has been able to solve it yet. It goes as follows:
- Show that if the equation $$phi(x) = n$$ has one solution; it always has a second solution, $n$ being given and $x$ being the
unknown.
Oddly enough, Carmichael didn't consider that this question deserved a star... In case you want to learn more about the history of this problem, I recommend that you take a look at the following installment of The evidence (a column that Stan Wagon used to contribute to The Mathematical Intelligencer):
S. Wagon, Carmichael's "empirical theorem". Math. Intelligencer, 8 (1986), No. 2, pp. 61-63.
edited yesterday
community wiki
José Hdz. Stgo.
1
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
yesterday
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
yesterday
2
$begingroup$
@user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
23 hours ago
$begingroup$
Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory.
$endgroup$
– Gerry Myerson
23 hours ago
add a comment |
1
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
yesterday
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
yesterday
2
$begingroup$
@user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
23 hours ago
$begingroup$
Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory.
$endgroup$
– Gerry Myerson
23 hours ago
1
1
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
yesterday
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
yesterday
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
yesterday
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
yesterday
2
2
$begingroup$
@user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
@user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
23 hours ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
23 hours ago
$begingroup$
Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory.
$endgroup$
– Gerry Myerson
23 hours ago
$begingroup$
Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory.
$endgroup$
– Gerry Myerson
23 hours ago
add a comment |
$begingroup$
The book "Théorie des nombres, Tome premier" by Edouard Lucas was published 1891. Many of the "Exemples" are actually exercises left to the reader. A scan is freely available in the archive.
$endgroup$
add a comment |
$begingroup$
The book "Théorie des nombres, Tome premier" by Edouard Lucas was published 1891. Many of the "Exemples" are actually exercises left to the reader. A scan is freely available in the archive.
$endgroup$
add a comment |
$begingroup$
The book "Théorie des nombres, Tome premier" by Edouard Lucas was published 1891. Many of the "Exemples" are actually exercises left to the reader. A scan is freely available in the archive.
$endgroup$
The book "Théorie des nombres, Tome premier" by Edouard Lucas was published 1891. Many of the "Exemples" are actually exercises left to the reader. A scan is freely available in the archive.
answered yesterday
community wiki
EFinat-S
add a comment |
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2
$begingroup$
Elementary Number Theory by Uspensky and Heaslet has a bunch of problems in each chapter. Not sure whether it's the oldness you are looking for (my impression is that the exercises don't get better if you go older than this).
$endgroup$
– darij grinberg
yesterday
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
yesterday
2
$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
yesterday
$begingroup$
Do you mean "unsolved" within the book itself, or "unsolved" anywhere?
$endgroup$
– James
17 hours ago