Reference request: Oldest number theory books with (unsolved) exercises? The 2019 Stack Overflow Developer Survey Results Are InClassical Enumerative Geometry ReferencesHow does “modern” number theory contribute to further understanding of $mathbbN$?Divergent Series as a topic of researchGeometric intuition for Fontaine-Wintenberger?Classification of singularities of plane curves of fixed degree (reference request)Reference request: Oldest calculus, real analysis books with exercises?Reference request: Oldest linear algebra books with exercises?Reference request for bounds of $n$-th compositeReference request: Oldest complex analysis books with (unsolved) exercises?Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?
Reference request: Oldest number theory books with (unsolved) exercises?
The 2019 Stack Overflow Developer Survey Results Are InClassical Enumerative Geometry ReferencesHow does “modern” number theory contribute to further understanding of $mathbbN$?Divergent Series as a topic of researchGeometric intuition for Fontaine-Wintenberger?Classification of singularities of plane curves of fixed degree (reference request)Reference request: Oldest calculus, real analysis books with exercises?Reference request: Oldest linear algebra books with exercises?Reference request for bounds of $n$-th compositeReference request: Oldest complex analysis books with (unsolved) exercises?Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?
$begingroup$
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
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
$endgroup$
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
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
$endgroup$
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
12 hours ago
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
12 hours ago
1
$begingroup$
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
6 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
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
$endgroup$
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
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
edited 12 hours ago
Get Off The Internet
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
asked 12 hours ago
Get Off The InternetGet Off The Internet
379320
379320
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
12 hours ago
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
12 hours ago
1
$begingroup$
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
6 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
12 hours ago
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
12 hours ago
1
$begingroup$
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
6 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
12 hours ago
$begingroup$
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
12 hours ago
1
1
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
12 hours ago
$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
12 hours ago
1
1
$begingroup$
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
6 hours ago
$begingroup$
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
6 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.
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
$endgroup$
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
8 hours ago
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
51 mins ago
1
$begingroup$
@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
9 mins ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
8 mins 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
5 mins 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.
answered 5 hours ago
EFinat-SEFinat-S
1,2241417
$endgroup$
add a comment |
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2 Answers
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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.
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
$endgroup$
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
8 hours ago
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
51 mins ago
1
$begingroup$
@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
9 mins ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
8 mins 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
5 mins 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.
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
$endgroup$
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
8 hours ago
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
51 mins ago
1
$begingroup$
@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
9 mins ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
8 mins 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
5 mins 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.
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
$endgroup$
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.
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
edited 6 hours ago
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
answered 9 hours ago
José Hdz. Stgo.José Hdz. Stgo.
5,36734878
5,36734878
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
8 hours ago
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
51 mins ago
1
$begingroup$
@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
9 mins ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
8 mins 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
5 mins 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
8 hours ago
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
51 mins ago
1
$begingroup$
@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
9 mins ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
8 mins 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
5 mins 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
8 hours ago
$begingroup$
Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
$endgroup$
– Gerry Myerson
8 hours ago
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
51 mins ago
$begingroup$
Is it a well-known problem that many number theory experts have seriously tried but failed to solve?
$endgroup$
– user21820
51 mins ago
1
1
$begingroup$
@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
9 mins 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
9 mins ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
8 mins ago
$begingroup$
@GerryMyerson: Okay thanks for the information!
$endgroup$
– user21820
8 mins 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
5 mins 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
5 mins 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.
answered 5 hours ago
EFinat-SEFinat-S
1,2241417
$endgroup$
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.
answered 5 hours ago
EFinat-SEFinat-S
1,2241417
$endgroup$
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.
answered 5 hours ago
EFinat-SEFinat-S
1,2241417
$endgroup$
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 5 hours ago
EFinat-SEFinat-S
1,2241417
answered 5 hours ago
EFinat-SEFinat-S
1,2241417
answered 5 hours ago
EFinat-SEFinat-S
1,2241417
answered 5 hours ago
EFinat-SEFinat-S
1,2241417
1,2241417
add a comment |
add a comment |
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2
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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).
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– darij grinberg
12 hours ago
1
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Not sure if there are exercises: books.google.com/books/about/…
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– Cherng-tiao Perng
12 hours ago
1
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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
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– EFinat-S
6 hours ago