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Research Activity

His Web Page

Field of Study

Number Theory

Subject of Study

Algebraic stucture of number fields (unit group, class group, diophantine equation) (Here the algebraic structure of algebraic number fields means the structure of class groups and unit groups of the fields, which are finite extensions of the rational field. Now we are mainly researching the following problems. First problem is the investigations of diophantine equations related to unit groups. Second one is the constructions of the finfinite family of algebraic number fields with explicit unit groups. We also concern the class number one problems of such infinite family of algebraic number fields and the abc conjecture which is one of the deepist conjecture of number theory.)

Book / Paper

Book:

1. Shin-ichi Katayama :
History of Science at Tokushima,
徳島科学史研究会, Tokushima, Dec. 2018.
2. Shin-ichi Katayama :
Proceedings of the 2003 Nagoya conference Yokoi-Chowla conjecture and related problems,
Furukawa Total Printing, Saga, Mar. 2004.

Academic Paper (Judged Full Paper):

1. Shin-ichi Katayama, Toru Nakahara, Nadia Khan and Hiroshi Sekiguchi :
The Gauß sum and its applications to number theory,
Journal of Basic & Applied Sciences, Vol.14, 230-234, 2018.
(DOI: 10.6000/1927-5129.2018.14.35)
2. Shin-ichi Katayama, Toru Nakahara, Nadia Khan and Takeshi Uehara :
Monogenity of totally real algebraic extension fields over a cyclotomic fields,
Journal of Number Theory, Vol.158, 348-355, 2016.
(DOI: 10.1016/j.jnt.2015.06.018,   Elsevier: Scopus)
3. Shin-ichi Katayama :
On Zenkoku Shijyo Suugaku Danwakai and Shikoku Suugaku Shijyo Danwa,
Natural Science Research, Faculty of Integrated Arts and Sciences, The University of Tokushima, Vol.28, No.4, 37-43, 2014.
(Tokushima University Institutional Repository: 106383)
4. Shin-ichi Katayama, Toru Nakahara, Ali Syed Inaya Shah, Mohammad Khalid and Sareer Badhah :
Orbit of quadratic irrationals molulo p by the modular group,
Journal of Research, Vol.25, No.1, 1-5, 2009.
5. Shin-ichi Katayama and Yasuhiro Kishi :
Infinite family of imaginary cyclic fields of degree p-1 with the p-rank of the ideal class groups of at least two,
Tsinghua Science & Technology, Vol.12, No.4, 475-478, 2007.
(DOI: 10.1016/S1007-0214(07)70070-6,   Elsevier: Scopus)
6. Shin-ichi Katayama, Claude Levesque and Toru Nakahara :
On a family of real bicyclic biquadartic fields,
CRM Proceedings and Lecture Notes, Vol.36, 165-174, 2004.
7. Shin-ichi Katayama and Claude Levesque :
On simultaneous diophantine equations,
Acta Arithmetica, Vol.108, No.4, 369-377, 2003.
(DOI: 10.4064/aa108-4-6,   Elsevier: Scopus)
8. Shin-ichi Katayama :
On zeta functions associated to finite groups,
Advanced Studies in Contemporary Mathematics, Vol.4, No.2, 111-125, 2002.
9. Shin-ichi Katayama, Claude Levesque and Toru Nakahara :
On the unit group and the class number of certain composita of two real quadratic fields,
Manuscripta Mathematica, Vol.105, No.1, 85-101, 2001.
(DOI: 10.1007/s002290170011,   Elsevier: Scopus)
10. Shin-ichi Katayama :
The abc conjecture, fundamental units and the simultaneous Pell equations,
Proceedings of the Jangjeon Mathematical Society, Vol.1, 19-26, 2000.
11. Shin-ichi Katayama :
The abc conjecture and the fundamental system of units of certain real bicyclic biquadratic fields,
Proceedings of the Japan Academy. Series A, Mathematical Sciences, Vol.75A, No.10, 198-199, 2000.
(DOI: 10.3792/pjaa.75.198,   Elsevier: Scopus)
12. Shin-ichi Katayama, Claude Levesque and Toru Nakahara :
On the units and class numbers of certain composita of two quadratic fields,
Proceedings of the Japan Academy, Vol.75A, No.4, 63-66, 1999.
13. Shin-ichi Katayama :
Real quadratic fields of Richaud-Degert type and related topics,
Advanced Studies in Contemporary Mathematics, Vol.1, 17-33, 1999.
14. Shin-ichi Katayama :
Remark on upper bounds for L(1,/chi),
Proceedings of the Japan Academy, Vol.72A, No.4, 89-90, 1996.
15. Shin-ichi Katayama and Yoko Kikuchi :
Some remarks on Egyptian fractions,
Mathematica Japonica, Vol.42, No.1, 127-130, 1995.
16. Shin-ichi Katayama and Shigeru Katayama :
Fibonacci, Lucas and Pell numbers and class numbers of bicyclic biquadratic fields,
Mathematica Japonica, Vol.42, No.1, 121-126, 1995.
17. Shin-ichi Katayama and Shigeru Katayama :
On bound functions of fundamental units of real quadratic fields,
Journal of Number Theory, Vol.46, No.3, 385-390, 1994.
18. Shin-ichi Katayama :
Group rings and the norm groups,
Proceedings of the Japan Academy, Vol.69A, No.9, 363-367, 1993.
19. Shin-ichi Katayama :
On fundamental units of real quadratic fields with norm +1,
Proceedings of the Japan Academy, Vol.68A, No.1, 18-20, 1992.
20. Shin-ichi Katayama :
On fundamental units of real quadratic fields with norm -1,
Proceedings of the Japan Academy, Vol.67A, No.10, 343-345, 1992.
21. Shin-ichi Katayama :
Isogenous tori and the class number formulae,
Journal of Mathematics of Kyoto University, Vol.31, No.3, 679-694, 1991.
22. Shin-ichi Katayama and Shigeru Katayama :
A note on the problem of Yokoi,
Proceedings of the Japan Academy, Vol.67A, No.1, 26-28, 1991.
23. Shin-ichi Katayama :
On products of consecutive integers,
Proceedings of the Japan Academy, Vol.66A, No.10, 305-306, 1991.
24. Shin-ichi Katayama and Shigeru Katayama :
On certain real quadratic fields with class numbers 3 and 5,
Proceedings of the Japan Academy, Vol.66A, No.9, 300-303, 1990.
25. Shin-ichi Katayama :
E(K/k) and other arithmetical invariants for finite Galois extensions,
Nagoya Mathematical Journal, Vol.114, 135-142, 1989.
26. Shin-ichi Katayama :
On unit groups of algebraic number fields,
Proceedings of the Japan Academy, Vol.64A, No.9, 345-346, 1988.
27. Shin-ichi Katayama :
The Euler number and other arithmetical invariants for finite Galois extensions of algebraic number fields,
Proceedings of the Japan Academy, Vol.63A, No.2, 27-30, 1987.
28. Shin-ichi Katayama :
Class number relations of algebraic tori II,
Proceedings of the Japan Academy, Vol.62A, No.8, 320-322, 1986.
29. Shin-ichi Katayama :
Class number relations of algebraic tori I,
Proceedings of the Japan Academy, Vol.62A, No.6, 216-218, 1986.
30. Shin-ichi Katayama :
A theorem on the cohomology of groups and some arithmetical applications,
Journal of Mathematics of Kyoto University, Vol.25, No.2, 365-391, 1985.
31. Shin-ichi Katayama :
On homotopy classes of cochain maps,
Proceedings of the Japan Academy, Vol.61A, No.5, 147-148, 1985.
32. Shin-ichi Katayama :
On the Tamagawa number of algebraic tori,
数学, Vol.37, No.1, 81-83, 1985.
33. Shin-ichi Katayama :
Cohomology groups of the unit group of a local field,
Proceedings of the Japan Academy, Vol.58A, No.9, 406-409, 1982.
34. Shin-ichi Katayama :
On the Galois cohomology groups of CK / DK,
Japanese Journal of Mathematics, Vol.8, No.2, 407-415, 1982.
35. Shin-ichi Katayama :
On the Galois cohomology groups of CK / DK,
Proceedings of the Japan Academy, Vol.57A, No.7, 378-380, 1981.

Academic Paper (Unrefereed Paper):

1. Shin-ichi Katayama and Hiroharu Yasui :
Goishi Hiroi and Fibonacci Number,
徳島科学史雑誌, Vol.38, 20-26, 2019.
(Tokushima University Institutional Repository: 115323)
2. Shin-ichi Katayama and Tomoya Kubo :
Wythoffの石取りゲームとRayleighの定理,
徳島科学史雑誌, Vol.37, 17-26, 2018.
(Tokushima University Institutional Repository: 113293,   CiNii: 120006629993)
3. Shin-ichi Katayama and Tomoya Kubo :
On Restricted Wythoff's Nim,
Journal of Mathematics, The University of Tokushima, Vol.52, 53-57, 2018.
4. Shin-ichi Katayama and Khan Nadia :
The abc Conjecture and Square Free Parts of Fibonacci Numbers,
Journal of Mathematics, The University of Tokushima, Vol.51, 5-28, 2017.
(Tokushima University Institutional Repository: 111822)
5. Shin-ichi Katayama :
石原侑氏の数学史論文について,
徳島科学史雑誌, Vol.35, 63-65, 2016.
6. Shin-ichi Katayama :
Hypothesis of Schinzel and Sierpinski and Cyclotomic Fields with Isomorphic Galois Groups,
Journal of Mathematics, The University of Tokushima, Vol.50, 43-47, 2016.
(Tokushima University Institutional Repository: 110912)
7. Shin-ichi Katayama :
Products of Arithmetic Progressions which are Squares,
Journal of Mathematics, The University of Tokushima, Vol.49, 9-12, 2015.
(Tokushima University Institutional Repository: 110904)
8. Shin-ichi Katayama :
Diophantine Equations and Hilbert's Theorem90,
Journal of Mathematics, The University of Tokushima, Vol.48, 37-42, 2014.
(Tokushima University Institutional Repository: 106352)
9. Shin-ichi Katayama :
フィボナッチの円周率の計算について,
徳島科学史雑誌, Vol.32, 1-8, 2013.
10. Shin-ichi Katayama :
Modified Farey trees and Pythagorean triples,
Journal of Mathematics, The University of Tokushima, Vol.47, 1-13, 2013.
(Tokushima University Institutional Repository: 105968,   CiNii: 120005352806)
11. Shin-ichi Katayama :
Scissors congruence for certain k-polygons,
Journal of Mathematics, The University of Tokushima, Vol.46, 1-12, 2012.
(Tokushima University Institutional Repository: 104983,   CiNii: 110009514554)
12. Shin-ichi Katayama :
Generalized Goggins's formula for Lucas and Companion Lucas sequences,
Journal of Mathematics, The University of Tokushima, Vol.45, 19-25, 2011.
(Tokushima University Institutional Repository: 84169,   CiNii: 110008729447)
13. Shin-ichi Katayama and Takuya Doi :
On partitions and k-polygons,
Journal of Mathematics, The University of Tokushima, Vol.44, 13-25, 2010.
(Tokushima University Institutional Repository: 72915,   CiNii: 110007870530)
14. Shin-ichi Katayama and Arata Kuroki :
A variation of Takagi's proof for quadratic reciprocity laws for Jacobi symbols,
Journal of Mathematics, The University of Tokushima, Vol.43, 9-23, 2009.
(Tokushima University Institutional Repository: 65208,   CiNii: 110007492238)
15. Shin-ichi Katayama and Takuya Doi :
A remark on partitions and triangles,
Journal of Mathematics, The University of Tokushima, Vol.43, 1-7, 2009.
(Tokushima University Institutional Repository: 65207,   CiNii: 110007492212)
16. Shin-ichi Katayama :
On some formulas for π/2,
Journal of Mathematics, The University of Tokushima, Vol.42, 13-17, 2008.
(Tokushima University Institutional Repository: 62141,   CiNii: 110007127172)
17. Shin-ichi Katayama :
Some infinite series of Fibonacci numbers,
Journal of Mathematics, The University of Tokushima, Vol.42, 9-12, 2008.
(Tokushima University Institutional Repository: 62140,   CiNii: 110007127173)
18. Shin-ichi Katayama :
On finite simple groups of cube order,
Journal of Mathematics, The University of Tokushima, Vol.41, No.0, 25-31, 2007.
(Tokushima University Institutional Repository: 57945,   CiNii: 110006853379)
19. Shin-ichi Katayama :
On the diophantine equation (x^2+1)(y^2+1)=(z^2+1)^2,
Journal of Mathematics, The University of Tokushima, Vol.40, 9-14, 2006.
(Tokushima University Institutional Repository: 53568,   CiNii: 110004849832)
20. Shin-ichi Katayama :
Maps R:Z/nZ → Z/n^2Z and some cryptographic applications,
Journal of Mathematics, The University of Tokushima, Vol.39, 7-21, 2005.
(Tokushima University Institutional Repository: 49023)
21. Shin-ichi Katayama :
A conjecture on fundamental units of real quadratic fields,
Journal of Mathematics, The University of Tokushima, Vol.35, 9-15, 2002.
(Tokushima University Institutional Repository: 141)
22. Shin-ichi Katayama :
Zeta functions of finite groups and arithmetically equivalent fields,
Journal of Mathematics, The University of Tokushima, Vol.35, 1-8, 2002.
(Tokushima University Institutional Repository: 140)
23. Shin-ichi Katayama :
Several methods for solving simultaneous Fermat-Pell equations,
Journal of Mathematics Tokushima University, Vol.33, 1-14, 2000.
(Tokushima University Institutional Repository: 133)
24. Shin-ichi Katayama :
On the class numbers of real quadratic fields of Richaud-Degert type,
Journal of Mathematics Tokushima University, Vol.31, 1-6, 1998.
(Tokushima University Institutional Repository: 120)
25. Shin-ichi Katayama :
Unit groups of some quartic fields,
Journal of Mathematics Tokushima University, Vol.28, 1-4, 1995.
(Tokushima University Institutional Repository: 103)
26. Shin-ichi Katayama and Shigeru Katayama :
On certain real bicyclic biquadratic fields with class number one and two,
Journal of Mathematics Tokushima University, Vol.26, 1-8, 1993.
(Tokushima University Institutional Repository: 95)
27. Shin-ichi Katayama and Kenji Kashihara :
On the structure of the integer solutions of z^2=(x^2-1)(y^2-1)+a,
Journal of Mathematics Tokushima University, Vol.24, 1-11, 1991.
(Tokushima University Institutional Repository: 89)

Review, Commentary:

1. Shin-ichi Katayama :
小出由岐太の求玉道間数法,
徳島新聞, Jul. 2017.
2. Shin-ichi Katayama :
海野十三と虫食い算,
徳島新聞, May 2015.
3. Shin-ichi Katayama :
海野十三と暗号,
徳島新聞, May 2014.
4. Shin-ichi Katayama :
マルコフ数の問題,
数学セミナー, Vol.52, No.52, 18-24, Jul. 2013.
(CiNii: 40019703128)

Proceeding of International Conference:

1. Shin-ichi Katayama :
On finite simple groups of power order,
Hawaii Conference in Algebraic Number Theory, Arithmetic Geometry, and Modular Forms, Honolulu, Mar. 2012.
2. Shin-ichi Katayama :
On finite simple groups of square order,
1078th American Mathematical Society Sectional Meeting, Honolulu, Mar. 2012.
3. Shin-ichi Katayama :
On a family of simultaneous Pell equations,
The seventh Japan-Korea Number Theory Symposium, Kujyu, Oct. 2004.
4. Shin-ichi Katayama and Yasuhiro Kishi :
An infinite family of imaginary cyclic fields of degree p-1 which have ideal class groups of p-ranks greater than one,
Yokoi-Chowla conjecture and related problems, Nagoya, Oct. 2003.
5. Shin-ichi Katayama :
Zeta functions of finite groups and arithmetically equivalent fields,
Seaway Number Theory Conference, Otawa (Canada), May 2001.
6. Shin-ichi Katayama :
The abc conjecture, fundamental units and the simultaneous Pell equations,
The fifth Japan-Korea Number Theory Symposium, Sendai, Nov. 1999.
7. Shin-ichi Katayama :
Units of certain quartic fields and integer points on an elliptic curve,
Workshop on Number Theory, Tokyo, Jul. 1999.
8. Shin-ichi Katayama :
Squares in Fibonacci sequence and unit indices of the biquadratic fields,
3rd Japan-Korea Symposium on Number Theory, Pohang(Korea), Jul. 1997.
9. Shin-ichi Katayama :
On composite real quadratic fields with class number one,
2nd Japan-Korea Symposium on Number Theory, Saga, Dec. 1996.
10. Shin-ichi Katayama :
Real quadratic fields of Richaud-Degert type and related topics,
1st Japan-Korea Symposium on Number Theory, Hapchun(Korea), Jul. 1996.

Proceeding of Domestic Conference:

1. Shin-ichi Katayama and 小山 優也 :
Corner the Knight Game,
日本フィボナッチ協会/研究集会, Aug. 2020.
2. Shin-ichi Katayama, 古家 尚樹 and 西岡 佑基 :
五角平方三角数,
日本フィボナッチ協会/研究集会, Aug. 2020.
3. Shin-ichi Katayama :
碁石ひろいゲームと数学,
日本科学史学会第23回西日本研究大会, Nov. 2019.
4. Shin-ichi Katayama and 安井 宏晴 :
碁石ひろいとフィボナッチ数,
日本フィボナッチ協会/研究集会, Aug. 2019.
5. Shin-ichi Katayama and 久保 智哉 :
制限付きのWythoffの二山崩し2,
日本フィボナッチ協会/研究集会, Aug. 2018.
6. Shin-ichi Katayama and 久保 智哉 :
取る石の数に制限のあるWythoffの二山崩し,
日本フィボナッチ協会/研究集会, Aug. 2017.
7. Shin-ichi Katayama, Kosuke Ono, Ayaka Sumi, Akira Kondoh and Kohei Matsuzawa :
海野十三と暗号,虫食い算,
海野十三忌2016, May 2016.
8. Shin-ichi Katayama, Kosuke Ono, Ayaka Sumi, Akira Kondoh and Kohei Matsuzawa :
海野十三と数学パズル,
日本科学史学会第19回西日本研究集会, Nov. 2015.
9. Shin-ichi Katayama :
NSW素数について,
日本フィボナッチ協会/研究集会, Aug. 2015.
10. Shin-ichi Katayama :
フィボナッチの著作と円周率計算,
日本フィボナッチ協会/研究集会, Aug. 2014.
11. Shin-ichi Katayama :
海野十三の暗号小説について,
日本科学史学会四国支部総会, Aug. 2014.
12. Shin-ichi Katayama :
フィボナッチの円周率の計算について,
日本科学史学会四国支部総会, Aug. 2013.
13. Shin-ichi Katayama :
On finite simple groups of power order,
Workshop in Number Theory in Saga 2011, Jan. 2011.
14. Shin-ichi Katayama :
RDタイプの実2次体のある種の分布と類数問題,
Workshop in Number Theory in Saga 2010, Jan. 2010.
15. Shin-ichi Katayama :
Class number problem of quadratic fields,
Galois theory and related problems Kanazawa 2009, Dec. 2009.
16. Shin-ichi Katayama :
Two topics on Gauss's half system,
Number Theory in Saga 2009, Jan. 2009.
17. Shin-ichi Katayama :
RDタイプの実2次体のある種の分布について,
Galois theory and related problems Tokushima 2008, Sep. 2008.
18. Shin-ichi Katayama and Hiroharu Yasui :
RSA Signatures and Redundancy,
10th Algorithm on Number Theory and Its Applications, May 2003.
19. Shin-ichi Katayama :
ABC conjecture and the fundamental system of units of certain algebraic number fields,
Inverse Galois problem and number theory, Aug. 1999.
20. Shin-ichi Katayama :
Group rings and the norm groups of cyclic extensions,
Nagoya Number Theory Seminar Spring Colloquim, Apr. 1993.

Report:

1. Shin-ichi Katayama :
Class number problem of quadratic fields,
Proceedings of the 8th Workshop on Number Theory, 81-96, Sep. 2010.
2. Shin-ichi Katayama :
On a family of simultaneous Pell equations,
Proceedings of the Japan-Korea Joint Seminar on Number Theory, 86-91, Apr. 2005.
3. Shin-ichi Katayama :
On class number one composita of real quadratic fields of Chowla-Yokoi type,
Proceedings of the 2003 conference Yokoi-Chowla conjecture and related problems, 51-58, Saga, Mar. 2004.
4. Shin-ichi Katayama, Claude Levesque and Toru Nakahara :
On the unit groups of certain quartic fields and simultaneous Fermat-Pell equations,
津田塾大学 数学·計算機科学研究所 研究所報, Vol.17, 73-82, Tokyo, Feb. 1999.
5. Shin-ichi Katayama :
On the class numbers od algebraic tori,
Proceedings of the 33th symposium on algebra, 1-12, Fukui, Jul. 1987.
6. Shin-ichi Katayama :
On the Galois cohomology groups of CK / DK,
RIMS Kokyuroku, Vol.440, 2-10, Kyoto, Jun. 1981.

Grants-in-Aid for Scientific Research (KAKEN Grants Database @ NII.ac.jp)

  • Number Theory and its Development to Discrete Mathematics (Project/Area Number: 20540019 )
  • Number Theory, Its Application to Discrete Mathematics and Development (Project/Area Number: 18540040 )
  • Number Theory and Its Applications to Discrete Mathematics (Project/Area Number: 16540029 )
  • Number Theory and Applications to the Related Discrete Mathematics (Project/Area Number: 14540033 )
  • ABC Conjecture and the Structure of Algebraic Number Fields (Project/Area Number: 14540030 )
  • Number Theory and Its Apprication to Discrete Mathematics (Project/Area Number: 11640036 )
  • 類数問題とHomology代数的構造の研究 (Project/Area Number: 07640053 )
  • シンプレクティック構造と数理物理学の研究 (Project/Area Number: 05640119 )
  • 可換環のホモロジー代数的性質の研究 (Project/Area Number: 05640048 )
  • 超函数と数理物理学の研究 (Project/Area Number: 04640165 )
  • 数理物理学とスペクトル理論の研究 (Project/Area Number: 03640163 )
  • Search by Researcher Number (70194777)