Skip to main content
MathOverflow

Return to Answer

Added a detail
Source Link
Salvo Tringali
Salvo Tringali
  • 10.5k
  • 2
  • 29
  • 64

This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):

  1. P. 165 in: W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).
  2. Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).
  3. Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 (click).
  4. Corollary 2: in: P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 (click).
  5. Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.
  6. Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.
  7. Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.
  8. Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 (click).

For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain property of densities.

This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):

  1. W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).
  2. Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).
  3. Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 (click).
  4. Corollary 2: in P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 (click).
  5. Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.
  6. Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.
  7. Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.
  8. Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 (click).

For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain property of densities.

This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):

  1. P. 165 in: W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).
  2. Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).
  3. Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 (click).
  4. Corollary 2 in: P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 (click).
  5. Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.
  6. Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.
  7. Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.
  8. Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 (click).

For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain property of densities.

added 10 characters in body
Source Link
Salvo Tringali
Salvo Tringali
  • 10.5k
  • 2
  • 29
  • 64

This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):

  1. W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).
  2. Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).
  3. Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 ([url=http://www.jstor.org/stable/2324814?seq=1#page_scan_tab_contents]click[/url]click).
  4. Corollary 2: in P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 ([url=http://www.jstor.org/stable/2974957?seq=1#page_scan_tab_contents]click[/url]click).
  5. Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.
  6. Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.
  7. Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.
  8. Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 ([url=http://www.jstor.org/stable/10.4169/amer.math.monthly.118.08.704]click[/url]click).[/quote]

For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain propertiesproperty of densities.

This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):

  1. W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).
  2. Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).
  3. Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 ([url=http://www.jstor.org/stable/2324814?seq=1#page_scan_tab_contents]click[/url]).
  4. Corollary 2: in P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 ([url=http://www.jstor.org/stable/2974957?seq=1#page_scan_tab_contents]click[/url]).
  5. Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.
  6. Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.
  7. Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.
  8. Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 ([url=http://www.jstor.org/stable/10.4169/amer.math.monthly.118.08.704]click[/url]).[/quote]

For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain properties of densities.

This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):

  1. W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).
  2. Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).
  3. Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 (click).
  4. Corollary 2: in P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 (click).
  5. Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.
  6. Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.
  7. Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.
  8. Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 (click).

For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain property of densities.

Source Link
Salvo Tringali
Salvo Tringali
  • 10.5k
  • 2
  • 29
  • 64

This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is a (partial) list of references where it is mentioned or proved (to try to convince you that yes, it is really well-known):

  1. W. Sierpiński, Elementary Theory of Numbers, North-Holland Mathematical Library 31, North-Holland, Amsterdam, 1988 (2nd edition).
  2. Theorem 4 in: P. Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1989 (2nd edition).
  3. Theorem 4 in: D. Hobby and D. M. Silberger, Quotients of primes, Amer. Math. Monthly 100 (1993), No. 1, 50–52 ([url=http://www.jstor.org/stable/2324814?seq=1#page_scan_tab_contents]click[/url]).
  4. Corollary 2: in P. Starni, Answers to two questions concerning quotients of primes, Amer. Math. Monthly 102 (1995), No. 4, 347–349 ([url=http://www.jstor.org/stable/2974957?seq=1#page_scan_tab_contents]click[/url]).
  5. Exercise 4.19 in: B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, Boston, 2007.
  6. Exercise 218 in: J.-M. De Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, RI, 2007.
  7. Exercise 7, p. 107 in: P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009.
  8. Corollary 5 in: S. R. Garcia, V. Selhorst-Jones, D. E. Poore, and N. Simon, Quotient sets and Diophantine equations, Amer. Math. Monthly 118 (2011), No. 8, 704–711 ([url=http://www.jstor.org/stable/10.4169/amer.math.monthly.118.08.704]click[/url]).[/quote]

For the record: I compiled this list at some point, based on information drawn from an arXiv preprint (unfortunately, I didn't take note of either a link, the author(s), or the title of the preprint, so I don't know how to find it again), as the question popped up in relation to a certain properties of densities.