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On the finiteness of certain Rabinowitsch polynomials II
Journal of Number Theory 99 (2003) 219–221
http://www.elsevier.com/locate/jnt
On the finiteness of certain Rabinowitsch polynomials II$ Dongho Byeona,* and H.M. Starkb b
a School of Mathematical Sciences, Seoul National University, Seoul, South Korea Department of Mathematics, University of California, San Diego, CA, 92037, USA
Received 9 April 2002; revised 10 June 2002 Communicated by D. Goss
Abstract 2 form Let m be a positive integer and fm ðxÞ be a polynomial of the p ffiffiffiffi fm ðxÞ ¼ x þ x m: We call a polynomial fm ðxÞ a Rabinowitsch polynomial if for t ¼ ½ m and consecutive integers x ¼ x0 ; x0 þ 1; y; x0 þ t 1; j f ðxÞj is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials fm ðxÞ such that 1 þ 4m is square free. In this note, we shall remove the condition that 1 þ 4m is square free. r 2002 Elsevier Science (USA). All rights reserved.
1. Introduction Let m be a positive integer and fm ðxÞ be a polynomial of the form fm ðxÞ ¼ pffiffiffiffi x2 þ x m: We call a polynomial fm ðxÞ a Rabinowitsch polynomial if for t ¼ ½ m and consecutive integers x ¼ x0 ; x0 þ 1; y; x0 þ t 1; j f ðxÞj is either 1 or prime. In [1], we proved that every Rabonowitsch polynomial of the form fm ðxÞ ¼ x2 þ x m is one of the following types. (i) x2 þ x 2; (ii) x2 þ x t2 ; where t is 1 or a prime, (iii) x2 þ x ðt2 þ t þ nÞ; where tonpt and where jnj is 1 or jnj ¼ 2tþ1 3 is an odd prime. $
The first author was supported by KOSEF Research Fund (01-0701-01-5-2). *Corresponding author. E-mail addresses: [email protected] (D. Byeon), [email protected] (H.M. Stark).
0022-314X/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 6 3 - X
220
D. Byeon, H.M. Stark / Journal of Number Theory 99 (2003) 219–221
Using this and the Siegel–Brauer theorem, we showed that there are only finitely many Rabinowitch polynomials fm ðxÞ ¼ x2 þ x m such that 1 þ 4m is square free. In this short note, we shall remove the condition that 1 þ 4m is square free by proving the following proposition. Proposition. If fm ðxÞ ¼ x2 þ x m is a Rabinowitsch polynomial, then 1 þ 4m is square free except m ¼ 2: Thus, we immediately have the following theorem. Theorem. There are only finitely many Rabinowitsch polynomials fm ðxÞ ¼ x2 þ x m: Finally, we shall give the table of all 14 Rabinowitsch polynomials fm ðxÞ ¼ x2 þ x m with at most one possible exception. If we assume the generalized Riemann hypothesis is true, then there is no exception.
2. Proof of Proposition Let D ¼ 1 þ 4m: Suppose p2 jD for some prime p (which must be odd). We see that for any x p1 2 ðmod pÞ; we have p2 j fm ðxÞ ¼ 14½ð2x þ 1Þ2 D: Let D ¼ 1 þ 4t2 in type (ii). If 1 þ 4t2 ¼ r2 for some positive integer r; then 1 ¼ ðr 2tÞðr þ 2tÞ: But it is impossible. So D cannot be a square and D ¼ D0 p2 ; where D0 a1; 4: If D ¼ 2p2 or 3p2 ; then D 2 or 3 ðmod 4Þ: It is a contradiction to D 1 ðmod 4Þ: Thus, we have that D ¼ D0 p2 ; where D0 X5: Now D ¼ 1 þ 4t2 X5p2 implies tXp: Let D ¼ ð2t þ 1Þ2 þ 4n in type (iii). If ð2t þ 1Þ2 þ 4n ¼ r2 for some positive integer r; then 4n ¼ ðr 2t 1Þðr þ 2t þ 1Þ: We easily see that it is impossible for 2 jnj ¼ 1: Let n be an odd prime such that jnj ¼ 2tþ1 3 : Then n divides r and n divides 4n: But it is also impossible. So D cannot be a square. By a similar argument to type (ii), we have that D ¼ D0 p2 ; where D0 X5: Now D ¼ ð2t þ 1Þ2 þ 4nX5p2 also implies tXp: Therefore for types (ii) and (iii), there is an x p1 2 ðmod pÞ in the range of fx ¼ x0 ; x1 ; y; x0 þ t 1g and p2 j fm ðxÞ: Thus fm ðxÞ is not a Rabinowitsch polynomial and we proved the proposition.
3. Table From proposition and results in [1,2,4], we have Table 1 of all 14 Rabinowitsch polynomials fm ðxÞ ¼ x2 þ x m with at most one possible exception.
D. Byeon, H.M. Stark / Journal of Number Theory 99 (2003) 219–221
221
Table 1 m
t
n
x0
1(1) 2 3 4 7 9 13 25 43 49 73 103 169 283
1(1) 1 1 2 2 3 3 5 6 7 8 10 13 16
(1)
0 0 0 1 0 1 0 1 0 1 0 4 1 6
1 1 1 1 1 7 11
D ¼ 1 þ 4m 5 13 17 29 37 53 101 173 197 293 413* 677 1133*
Type ii(iii) i iii ii iii ii iii ii iii ii iii iii ii iii
Remark 1. While making Table 1, we know that Mollin and Williams [3] obtained a similar result to our previous work in [1], by a different method. They treated the case of x0 ¼ 1; square-free 1 þ 4m and produced narrow Richaud–Degert type of real quadratic fields associated to Rabinowitsch polynomials. But we considered the case of arbitrary x0 ; 1 þ 4m and obtained not only narrow Richaud–Degert type but also wide Richaud–Degert type of real quadratic fields (* in Table 1). Remark 2. Example (iii)-4 in [1] is not correct, because 1 þ 4m ¼ 245 ¼ 5 72 is not square free and 102 þ 10 61 ¼ 49 ¼ 72 is not prime. Note added in proof: S. Louboutin also found a proof of the proposition which is essentially the same as ours, and he kindly sent his proof to us after this paper was accepted. Acknowledgments The authors thank the referee for many helpful suggestions. References [1] D. Byeon, H.M. Stark, On the finiteness of certain Rabinowitsch polynomials, J. Number Theory 94 (2002) 177–180. [2] H.K. Kim, M.G. Leu, T. Ono, On two conjectures on Real quadratic fields, Proc. Japan Acad. Ser. A 63 (1987) 222–224. [3] R.A. Mollin, H.C. Williams, On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J. 112 (1988) 143–151. [4] R.A. Mollin, H.C. Williams, Prime producing quadratic polynomials and real quadratic fields of class number one, Theorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 654–663.
http://www.elsevier.com/locate/jnt
On the finiteness of certain Rabinowitsch polynomials II$ Dongho Byeona,* and H.M. Starkb b
a School of Mathematical Sciences, Seoul National University, Seoul, South Korea Department of Mathematics, University of California, San Diego, CA, 92037, USA
Received 9 April 2002; revised 10 June 2002 Communicated by D. Goss
Abstract 2 form Let m be a positive integer and fm ðxÞ be a polynomial of the p ffiffiffiffi fm ðxÞ ¼ x þ x m: We call a polynomial fm ðxÞ a Rabinowitsch polynomial if for t ¼ ½ m and consecutive integers x ¼ x0 ; x0 þ 1; y; x0 þ t 1; j f ðxÞj is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials fm ðxÞ such that 1 þ 4m is square free. In this note, we shall remove the condition that 1 þ 4m is square free. r 2002 Elsevier Science (USA). All rights reserved.
1. Introduction Let m be a positive integer and fm ðxÞ be a polynomial of the form fm ðxÞ ¼ pffiffiffiffi x2 þ x m: We call a polynomial fm ðxÞ a Rabinowitsch polynomial if for t ¼ ½ m and consecutive integers x ¼ x0 ; x0 þ 1; y; x0 þ t 1; j f ðxÞj is either 1 or prime. In [1], we proved that every Rabonowitsch polynomial of the form fm ðxÞ ¼ x2 þ x m is one of the following types. (i) x2 þ x 2; (ii) x2 þ x t2 ; where t is 1 or a prime, (iii) x2 þ x ðt2 þ t þ nÞ; where tonpt and where jnj is 1 or jnj ¼ 2tþ1 3 is an odd prime. $
The first author was supported by KOSEF Research Fund (01-0701-01-5-2). *Corresponding author. E-mail addresses: [email protected] (D. Byeon), [email protected] (H.M. Stark).
0022-314X/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 6 3 - X
220
D. Byeon, H.M. Stark / Journal of Number Theory 99 (2003) 219–221
Using this and the Siegel–Brauer theorem, we showed that there are only finitely many Rabinowitch polynomials fm ðxÞ ¼ x2 þ x m such that 1 þ 4m is square free. In this short note, we shall remove the condition that 1 þ 4m is square free by proving the following proposition. Proposition. If fm ðxÞ ¼ x2 þ x m is a Rabinowitsch polynomial, then 1 þ 4m is square free except m ¼ 2: Thus, we immediately have the following theorem. Theorem. There are only finitely many Rabinowitsch polynomials fm ðxÞ ¼ x2 þ x m: Finally, we shall give the table of all 14 Rabinowitsch polynomials fm ðxÞ ¼ x2 þ x m with at most one possible exception. If we assume the generalized Riemann hypothesis is true, then there is no exception.
2. Proof of Proposition Let D ¼ 1 þ 4m: Suppose p2 jD for some prime p (which must be odd). We see that for any x p1 2 ðmod pÞ; we have p2 j fm ðxÞ ¼ 14½ð2x þ 1Þ2 D: Let D ¼ 1 þ 4t2 in type (ii). If 1 þ 4t2 ¼ r2 for some positive integer r; then 1 ¼ ðr 2tÞðr þ 2tÞ: But it is impossible. So D cannot be a square and D ¼ D0 p2 ; where D0 a1; 4: If D ¼ 2p2 or 3p2 ; then D 2 or 3 ðmod 4Þ: It is a contradiction to D 1 ðmod 4Þ: Thus, we have that D ¼ D0 p2 ; where D0 X5: Now D ¼ 1 þ 4t2 X5p2 implies tXp: Let D ¼ ð2t þ 1Þ2 þ 4n in type (iii). If ð2t þ 1Þ2 þ 4n ¼ r2 for some positive integer r; then 4n ¼ ðr 2t 1Þðr þ 2t þ 1Þ: We easily see that it is impossible for 2 jnj ¼ 1: Let n be an odd prime such that jnj ¼ 2tþ1 3 : Then n divides r and n divides 4n: But it is also impossible. So D cannot be a square. By a similar argument to type (ii), we have that D ¼ D0 p2 ; where D0 X5: Now D ¼ ð2t þ 1Þ2 þ 4nX5p2 also implies tXp: Therefore for types (ii) and (iii), there is an x p1 2 ðmod pÞ in the range of fx ¼ x0 ; x1 ; y; x0 þ t 1g and p2 j fm ðxÞ: Thus fm ðxÞ is not a Rabinowitsch polynomial and we proved the proposition.
3. Table From proposition and results in [1,2,4], we have Table 1 of all 14 Rabinowitsch polynomials fm ðxÞ ¼ x2 þ x m with at most one possible exception.
D. Byeon, H.M. Stark / Journal of Number Theory 99 (2003) 219–221
221
Table 1 m
t
n
x0
1(1) 2 3 4 7 9 13 25 43 49 73 103 169 283
1(1) 1 1 2 2 3 3 5 6 7 8 10 13 16
(1)
0 0 0 1 0 1 0 1 0 1 0 4 1 6
1 1 1 1 1 7 11
D ¼ 1 þ 4m 5 13 17 29 37 53 101 173 197 293 413* 677 1133*
Type ii(iii) i iii ii iii ii iii ii iii ii iii iii ii iii
Remark 1. While making Table 1, we know that Mollin and Williams [3] obtained a similar result to our previous work in [1], by a different method. They treated the case of x0 ¼ 1; square-free 1 þ 4m and produced narrow Richaud–Degert type of real quadratic fields associated to Rabinowitsch polynomials. But we considered the case of arbitrary x0 ; 1 þ 4m and obtained not only narrow Richaud–Degert type but also wide Richaud–Degert type of real quadratic fields (* in Table 1). Remark 2. Example (iii)-4 in [1] is not correct, because 1 þ 4m ¼ 245 ¼ 5 72 is not square free and 102 þ 10 61 ¼ 49 ¼ 72 is not prime. Note added in proof: S. Louboutin also found a proof of the proposition which is essentially the same as ours, and he kindly sent his proof to us after this paper was accepted. Acknowledgments The authors thank the referee for many helpful suggestions. References [1] D. Byeon, H.M. Stark, On the finiteness of certain Rabinowitsch polynomials, J. Number Theory 94 (2002) 177–180. [2] H.K. Kim, M.G. Leu, T. Ono, On two conjectures on Real quadratic fields, Proc. Japan Acad. Ser. A 63 (1987) 222–224. [3] R.A. Mollin, H.C. Williams, On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J. 112 (1988) 143–151. [4] R.A. Mollin, H.C. Williams, Prime producing quadratic polynomials and real quadratic fields of class number one, Theorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 654–663.