Tokushima University / Educator and Researcher Directory --- Katayama, Shin-ichi

Personal Web Page

https://researchmap.jp/read0017358 (researchmap)

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. (Summary) 徳島新聞」に2013年4月から2018年9月まで月一回連載された「徳島の科学史」全66回をまとめた． (Keyword) 科学史
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. (Summary) Let K be the biqudratic field and L be a cyclic bicubic fild obtained by composition of the quadratic field of conductor 5 and a cubic field appliying cubic Gauss sum. Then we determine when K is monogenic and L is non-monohenic except for the cases when the Gauss sums are conductor 7 and 9. (Keyword) monogenic (Link to Publication Site) ● Publication site (DOI): 10.6000/1927-5129.2018.14.35 (Link to Search Site for Scientific Articles) ● Search Scopus @ Elsevier (DOI): 10.6000/1927-5129.2018.14.35 (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. (Summary) Let k be a cyclotomic field and F be a totally real algebraic field ocer the rational field Q. We have proved the composite field of k and F is not monogenic under certain conditions on the cyclotomic fields k. (Link to Publication Site) ● Publication site (DOI): 10.1016/j.jnt.2015.06.018 (Link to Search Site for Scientific Articles) ● Summary page in Scopus @ Elsevier: 2-s2.0-84940416734 (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. (Summary) Zenkoku Shijo Sugaku Danwakai is a Japanese mathematical journal published by the Department of Mathematics, Faculty of Sciences of Osaka University from 1934 to 1949. This mimeographed journal played an important role in the mathematical research announce ments during World War II. Recently this journal was uploaded as pdf files at the HP of the Department of Mathematics of Osaka University. The URL ishttp://www.math.sci.osaka-u.ac.jp/shijodanwakai/. Shikoku Sugaku Shijo Danwa is also a Japanesemathematicalmimeographed journal published by the Department of Mathematics, the Gakugei College of TokushimaUniversity from 1951 to 1956. We have compared the authors,titles and contents of these journals andconcluded that Shikoku Sugaku Shijo Danwa played a successive role to Zenkoku Shijo Sugaku Danwakai. In 2014, we have uploaded this mimeographed journal as pdf files at the HP of the Department of MathematicalSciencesof Tokushima University (http://www-math.ias.tokushima-u.ac.jp/). We must mention that manyfrontier resultsinthose daysareincluded in these journals,althoughthey were written in Japanes (Keyword) Mathematical Journal (Tokushima University Institutional Repository) ● Metadata: 106383 (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. (Summary) We gave a simple geometric explanations for the transformation of a quadratic irrational modulo p.
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. (Summary) Applying Imaoka-Kishi's construction of cyclic p-1 fields of class number divisible by p to Katayama's special bicyclic biquadratic fields, we have constructed infinitely many cyclic extensions of degree p-1 withthe p-ranks of the class groups st least 2. (Link to Publication Site) ● Publication site (DOI): 10.1016/S1007-0214(07)70070-6 (Link to Search Site for Scientific Articles) ● Summary page in Scopus @ Elsevier: 2-s2.0-34447619670 (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. (Summary) This is a continuation of our previous paper published in Manuscripta Mathematics(2001).
7.	Shin-ichi Katayama and Claude Levesque : On simultaneous diophantine equations, Acta Arithmetica, Vol.108, No.4, 369-377, 2003. (Summary) We generalized the results of Bennett and Walsh on the integer solutions of certain simultaneous Pell equations to certain family of simultaneous diophantine equations, using the structure of the unit groups of certain biquadratic fields. (Keyword) SIMULTANEOUS PELL EQUATIONS / ABC CONJECTURE / number / FIELDS (Link to Publication Site) ● Publication site (DOI): 10.4064/aa108-4-6 (Link to Search Site for Scientific Articles) ● Summary page in Scopus @ Elsevier: 2-s2.0-0042834167 (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. (Summary) We defined certain zeta functions associated to finite groups and consider the problem when two zeta functions are coincide. As aa application of this consideration, we have estimated the number of arithemetically equivalent fields for the special cases which we have constructed.
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. (Summary) This is a full paper of the announced results of Proc.Japan Acad. 75A(1999). (Link to Publication Site) ● Publication site (DOI): 10.1007/s002290170011 (Link to Search Site for Scientific Articles) ● Summary page in Scopus @ Elsevier: 2-s2.0-0035619182 (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. (Summary) This is a proceeding of our talk at the 5th Japan-Korea Number Theory symposium and is the first volume of this academic journal.
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. (Summary) We have shown that, under the ABC conjecture, we can generate the fundamental system of units of quartic fields which have been treated in Proc. Japan Acad. 75A(1999),63-66. (Link to Publication Site) ● Publication site (DOI): 10.3792/pjaa.75.198 (Link to Search Site for Scientific Articles) ● Summary page in Scopus @ Elsevier: 2-s2.0-22844456406 (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. (Summary) We proved the analogous results as Math.Japonica 42(1995) for the genral biciclic biquadartic fields associated to binary recurrence sequences with odd M.
13.	Shin-ichi Katayama : Real quadratic fields of Richaud-Degert type and related topics, Advanced Studies in Contemporary Mathematics, Vol.1, 17-33, 1999. (Summary) This is a survay paper on the class number problems including the real quadratic fields of Richaud-Degert type and several new results obtained in J.Math.Tokushima Univ. 31 (1998) are contained in this paper.
14.	Shin-ichi Katayama : Remark on upper bounds for L(1,/chi), Proceedings of the Japan Academy, Vol.72A, No.4, 89-90, 1996. (Summary) The special values of L-functions associated to Kronecker characters of real quadartic fields are important invariants related to the class numbers of real quadratic fields. Classical result of Hua was improved by Stanton, Sudler and Williams. Recently T.Ono obtained new upper estimate for the special values. We compared these 3 estimates and find the exact values where the order of the estimates changes.
15.	Shin-ichi Katayama and Yoko Kikuchi : Some remarks on Egyptian fractions, Mathematica Japonica, Vol.42, No.1, 127-130, 1995. (Summary) Egyptian fraction expansion is a classical diophantine equation which has its origin in "Rhind Mathematical Papyrus". In this paper, we calssifyed and generalized the formulas of Rhind Papyrus and considered certain zeta function related to the number of representaions of a unit fraction as the sum of two unit fractions.
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. (Summary) We considered some families of bicyclic biquadratic fields associated to Fibonacci numbers and Pell numbers. We determined the structure of unit groups and obtained the class number formulas explicitly.
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. (Summary) We introduced the notion of bound function to treat all the class number problems concerning real quadartic fields. As a colloary of our results, we have shown part of the problems of Yokoi can be reduced to the results of Mollin and Williams.
18.	Shin-ichi Katayama : Group rings and the norm groups, Proceedings of the Japan Academy, Vol.69A, No.9, 363-367, 1993. (Summary) We generalized Hirbert's Theorem 90 as the action of the integrated group ring on the multiplicative group of cyclic extensions.
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. (Summary) We descoverd new criterion for certain unit of norm +1 to be the fundamental unit of real quadratic fields.
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. (Summary) We have discovered new criterion for certain unit of norm -1 to be the fundamental unit of real quadratic fields.
21.	Shin-ichi Katayama : Isogenous tori and the class number formulae, Journal of Mathematics of Kyoto University, Vol.31, No.3, 679-694, 1991. (Summary) We have shown the class number formulae for the exact sequence of algebraic tori. As a colloraly, we have obtained a purely algebraic proof of the class number formula for relative bicyclic biquadratic fields, using Galois cohomology. The class number formula was now rediscoverd and generalized by F.Lemmermeyer.
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. (Summary) We determined all the real quadratic fields with class number one for certain inifinite family of real quadratic fields defined by H.Yokoi(under generalized Riemann Hypothesis).
23.	Shin-ichi Katayama : On products of consecutive integers, Proceedings of the Japan Academy, Vol.66A, No.10, 305-306, 1991. (Summary) Consider the problem when the products of the products of two consecutive integers are again the productts of two consectuive integers. We have shown all the infinite integer solutions of the diophantine equations x(x+1)y(y+1)=z(z+1) are obtained by the action of a natural group G* on the trivial solution (0,0,0).
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. (Summary) M.G.Leu and H.K.Kim and other mathematicians have determined all the quadratic fields of Yokoi-Chowla type of class number 1 and 2 under the Riemann hypothesis. We have determined all such fields with class number 3 and 5 under the same situation.
25.	Shin-ichi Katayama : E(K/k) and other arithmetical invariants for finite Galois extensions, Nagoya Mathematical Journal, Vol.114, 135-142, 1989. (Summary) This is the detailed report of the announcement published in Proceedings of the Japan Academy on 1987.
26.	Shin-ichi Katayama : On unit groups of algebraic number fields, Proceedings of the Japan Academy, Vol.64A, No.9, 345-346, 1988. (Summary) We have generalized our results on the certain residue groups of the unit groups of algebraic number fields.
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. (Summary) Recently, new arithmetical invariant E(K/k) has been defined and investigated by Takashi Ono. In this paper, we have shown the explixit relations between the genus number g(K/k) and other invariants and E(K/k) for any Galois extension K/k.
28.	Shin-ichi Katayama : Class number relations of algebraic tori II, Proceedings of the Japan Academy, Vol.62A, No.8, 320-322, 1986. (Summary) We generalized the results in "Class number relations of algebraic tori I" to non-galois extension K/k using I.T.Adamson's non-galois cohomology.
29.	Shin-ichi Katayama : Class number relations of algebraic tori I, Proceedings of the Japan Academy, Vol.62A, No.6, 216-218, 1986. (Summary) For any finite Galois extension K/k, T.Ono defined two arithmetical invariants E(K/k) and E'(K/k) using the Weil functor and algebraic tori. We have write down E(K/k) and E'(K/k) by the orders of galois cohomology groups explicitly.
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. (Summary) One can realize the Artin map in class field theory using the cannonical class of Galois cohomology groups. We generalize this map using the method of Tate on cohomologically trivial Galois moudules.
31.	Shin-ichi Katayama : On homotopy classes of cochain maps, Proceedings of the Japan Academy, Vol.61A, No.5, 147-148, 1985. (Summary) Let A and B be the cochain complexs. [A,B] denotes the abelian group of homotopy classes of cohain maps from A to B. E(A_#,B) denotes the abelian group of equivalent classes of weakly splitting extensions of A_# and B. Then [A,B] is isomorphic to E(A_#,B).
32.	Shin-ichi Katayama : On the Tamagawa number of algebraic tori, 数学, Vol.37, No.1, 81-83, 1985. (Summary) Over 20 years ago, T.Ono conjectured any positive rational number appears as the Tamagawa number of certain algebraic torus. We proved the conjecture affirmatively, reducing the problem to the construction certain abelian extensions with everywhere cyclic local extensions
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. (Summary) Let K/k be a local Galois extension. Let U_K be a local unit group and ar{Z} be the Artin's splitting module defined by the cannonial class of K/k. Then there exists an isomoriphism between the Galois cohomology groups of U_K and the Galois cohomology groups of ar{Z}.
34.	Shin-ichi Katayama : On the Galois cohomology groups of C_K / D_K, Japanese Journal of Mathematics, Vol.8, No.2, 407-415, 1982. (Summary) Let K/k be a finite Galois extension. C_K denotes the idele class group and D_K be the coonected component of the unity in C_K. We wrote down the Galois cohomology group of C_K/D_K using the 2-Sylow groups of Gal(K/k).
35.	Shin-ichi Katayama : On the Galois cohomology groups of C_K / D_K, Proceedings of the Japan Academy, Vol.57A, No.7, 378-380, 1981. (Summary) We announced part of the results in Japanese Journal of Mathematics 8 (407-415).

Academic Paper (Unrefereed Paper):

1.	Shin-ichi Katayama : Coner the Knight Game, Journal of Mathematics, Tokushima University, Vol.55, 83-88, 2021. (Summary) Wythoff's game is equivalent to a game the movements of Queen of the chess on the infinite chess board, which is named "Corner the Queen Game". In this paper, we have introduced a new game, where the queen is replaced by the knight and determined the Grundy numbers of this new game. (Keyword) Wythoff's NIm (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050009371656770432 (CiNii: 1050009371656770432)
2.	Shin-ichi Katayama : On Polygonal Square Triangular Numbers II, Journal of Mathematics, Tokushima University, Vol.55, 1-10, 2021. (Summary) This is a continuation of our previous paper "On polygonal square triangular numbers. We shall estimate the number of solutions of related simultaneous diophantine equations. We also find some example of polygonal square numbers(toridecagonal square triangular number) with exact 2 solutions(1 and 36). (Keyword) polygonal number / Integral points on modular elliptic curves (Tokushima University Institutional Repository) ● Metadata: 116691 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050572321610199040 (Tokushima University Institutional Repository: 116691, CiNii: 1050572321610199040)
3.	Shin-ichi Katayama, Akihito Sumitomo and Taiyoh Maruyama : A Report on the abc Conjecture, 徳島科学史雑誌, Vol.39, 19-27, 2020. (Summary) abc予想は，1985年前後にOesterleとMasserによって提唱された予想であり，2020年4月に京都大学数理科学研究所の望月新一氏が2012年から主張していた証明が数理研の査読誌に正式に受理された．本論文では，abc予想の成立から2012年の発表から2020年の京都大の発表までの歴史を概観した上で，強いabc予想からフェルマー・ワイルスの定理が導かれること，弱いabc予想からフェルマー・カタラン予想が導かれることを紹介している．さらにabc予想の定式化に表れるquarityの分布に関していくつかの計算結果と考察を新たに行いその結果を報告した． (Keyword) abc予想 / フェルマー予想
4.	Shin-ichi Katayama, Naoki Furuya and Yuki Nishioka : On Polygonal Square Triangular Numbers, Journal of Mathematics, The University of Tokushima, Vol.54, 1-12, 2020. (Summary) The proof of the fact that the pentagonal square triangular number is only 1 has not been published in any article or text. The authors give an elementary proof of this probably well known result. They proved more general problem which find the k-gonal square triangular numbers for small k(e.g. k=5, 7, 8, 9, 10, 11, 12) and several other cases and found there are no solutions other that 1 for these cases. They reduced these problems to the the determination of integer points on related elliptic curves over Q. (Tokushima University Institutional Repository) ● Metadata: 115943 (Tokushima University Institutional Repository: 115943)
5.	Shin-ichi Katayama and Yuya Koyama : Square, Rectangular and Triangular Nim Games, Journal of Mathematics, The University of Tokushima, Vol.54, 93-104, 2020. (Summary) This is a continuation of the first author's Restricted Wythoff's Nim Game. It is known that Wythoff's Nim is equivalent to the corner the queen game on the infinite chess board. Motivated by the recent research on Ryuoh and Ryuma Nim, we developed and investigated several games which generalize the case of gold general(of Shogi), which is called Square NIm. We also investigated Rectangular NIm(a genaralozation of Square Nim) and Triangular Nim and obtaied general results on these games. (Tokushima University Institutional Repository) ● Metadata: 115957 (Tokushima University Institutional Repository: 115957)
6.	Shin-ichi Katayama and Hiroharu Yasui : Goishi Hiroi and Fibonacci Number, 徳島科学史雑誌, Vol.38, 20-26, 2019. (Summary) 「碁石ひろい」は，和算での碁石に関するゲームである．最近碁石ひろいゲームは，NP完全なゲームであることが証明されているが，ここでは，取りきることができるある碁石の配置の無限系列とその取り尽くし可能な総数の下限がフィボナッチ数で記述できること証明した． (Keyword) 石取りゲーム / フィボナッチ数 (Tokushima University Institutional Repository) ● Metadata: 115323 (Tokushima University Institutional Repository: 115323)
7.	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. (Summary) 日本における明治以降の石取りゲームの受容史と，最近の石取りゲームに関する結果について概説した．特にWythoffの二山崩しに使われているRauleighの定理が日本では長く間違った名称で紹介されてきた経緯とRayleightの定理の出典について紹介し，最近著者によって得られた制限付きのWythoffの二山崩しに関する結果に関しても紹介した． (Keyword) 石取りゲーム / レイリーの定理 (Tokushima University Institutional Repository) ● Metadata: 113293 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050564288246558848 (Tokushima University Institutional Repository: 113293, CiNii: 1050564288246558848)
8.	Shin-ichi Katayama and Tomoya Kubo : On Restricted Wythoff's Nim, Journal of Mathematics, The University of Tokushima, Vol.52, 53-57, 2018. (Summary) In Game theory, extracting games of stones are described by the abelian group called Nim sum. In 1907, Wythoff modified Nim game and the winning strategy is related to Fibonacci numbers. Recently in 1999, Dress, Flamennkanp and Pink showed the grudy numbers of Wythoff's Nim are additive periodic. In this paper, we have restricted the number of stones can be picked up from one pile is finite and proved the grundy number is periodic for this case.t (Tokushima University Institutional Repository) ● Metadata: 112903 (Tokushima University Institutional Repository: 112903)
9.	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. (Summary) Under the abc conjecture, we have shown the finiteness of certain family of real quadratic fields related to Fibonacci numbers. In this paper, we have determined all the real quadratic fields in the above family which has the solutions related to Eisenstein's problem. (Tokushima University Institutional Repository) ● Metadata: 111822 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050282812443147648 (Tokushima University Institutional Repository: 111822, CiNii: 1050282812443147648)
10.	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. (Summary) 徳島科学史雑誌35号の石原侑氏の追悼文の一つとして，石原氏の2つの論文「小出長十郎『砲術玉道真法』について」(徳島科学史雑誌5号(1986))，および「ルカシェービッチ記法」(同7号(1988))の内容と筆者の最近の数学史研究の関係について紹介した． (Keyword) 砲術玉道真法 / ポーランド記法および逆ポーランド記法
11.	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. (Summary) We have generalized Carmichael conjecture, which states the existence of different m with φ(m)=φ(n) for any natural number n to the existence of cyclotomic extension Q(m) with isomorphic galois group Gal(Q(n)/Q). Under the Hypothesis of Schinzel and Sierpinski, we have shown the existence of certain pairs (m,n) with isomorphic Galois groups. (Tokushima University Institutional Repository) ● Metadata: 110912 (Tokushima University Institutional Repository: 110912)
12.	Shin-ichi Katayama : Products of Arithmetic Progressions which are Squares, Journal of Mathematics, The University of Tokushima, Vol.49, 9-12, 2015. (Summary) Let f(x) be the product of arithmetic progressions x(x-b)(x+b). Then It is a open probrem when f(x)f(y) is squre with nontrivial x and y. In this short remark, using the norm equations of Q(√2), we have proved there exist infinitely many nontrivial integer solutions. (Tokushima University Institutional Repository) ● Metadata: 110904 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050564287419650176 (Tokushima University Institutional Repository: 110904, CiNii: 1050564287419650176)
13.	Shin-ichi Katayama : Diophantine Equations and Hilbert's Theorem90, Journal of Mathematics, The University of Tokushima, Vol.48, 37-42, 2014. (Summary) Let K/F S be a cyclic field extension of degree n with the Galois group generated by t. Then Hilbert's Theorem 90 states that any element a in K with relative norem 1 can be expressed as b/b^t with some element b in K. We have pointed out there exists a family of diophantine equations whose rational solutions are obtained by applying Hilbert's Theorem 90, which include thw well known the parametrizations of the primitive Pythagorean triples. (Tokushima University Institutional Repository) ● Metadata: 106352 (Tokushima University Institutional Repository: 106352)
14.	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. (Summary) フィボナッチの残した2つの書籍「Liber Abachi」(1202)，「De Practica Geometrie」(1220)をもとに，フィボナッチの行った円周率計算を，その数学的な意味および位取り記数法の導入についてのフィボナッチの寄与を考慮に入れて．アルキメデスが行っていた円周率計算と比較検討した． (Keyword) 円周率 / Fibonacci
15.	Shin-ichi Katayama : Modified Farey trees and Pythagorean triples, Journal of Mathematics, The University of Tokushima, Vol.47, 1-13, 2013. (Summary) In 1963, F. J. M. Barning has dicovered a ternary tree of primitive Pythagorean triples. In this paper, we have given an elementary explanation of this fact using classical Euclidean parametrization of primitive Pythagorean trples and a new concept of modified Farey trees. (Tokushima University Institutional Repository) ● Metadata: 105968 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050282812441571456 (Tokushima University Institutional Repository: 105968, CiNii: 1050282812441571456)
16.	Shin-ichi Katayama : Scissors congruence for certain k-polygons, Journal of Mathematics, The University of Tokushima, Vol.46, 1-12, 2012. (Summary) Let S be the set of k-polygons where the vertices chosn from the n equally division points of the given circle. Then we have verified that the two notions scissors congruent and congruent do'nt coincide in the set S. But these notions are coincide for the special cases for k=3 and n=p (prime). (Tokushima University Institutional Repository) ● Metadata: 104983 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050564287418262784 (Tokushima University Institutional Repository: 104983, CiNii: 1050564287418262784)
17.	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. (Summary) We have genralized our two results which have published in Journal of mathematics, the university of Tokushima Vol. 42(2008) to Lucas and companion Lucas sequences. (Tokushima University Institutional Repository) ● Metadata: 84169 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050001337463838208 (Tokushima University Institutional Repository: 84169, CiNii: 1050001337463838208)
18.	Shin-ichi Katayama and Takuya Doi : On partitions and k-polygons, Journal of Mathematics, The University of Tokushima, Vol.44, 13-25, 2010. (Summary) Let C be a given circle divided into n parts equally. Let Ck (n) be the number of incongruent k-polygons inscribed in C, where the vertices of the triangles are chosen from the ends of these parts. In this paper, we have determined Ck(n) for k=4 an 5 and related generating functions. These results are the generalizations of our result for k=3(J. Math. Tokushima Univ. 43). (Tokushima University Institutional Repository) ● Metadata: 72915 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050845762393844736 (Tokushima University Institutional Repository: 72915, CiNii: 1050845762393844736)
19.	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. (Summary) In this paper, we have generalized Takagi's proof for quadratic reciprocity laws for Legendre symbols for Jacobi symbols. (Tokushima University Institutional Repository) ● Metadata: 65208 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050845762393831936 (Tokushima University Institutional Repository: 65208, CiNii: 1050845762393831936)
20.	Shin-ichi Katayama and Takuya Doi : A remark on partitions and triangles, Journal of Mathematics, The University of Tokushima, Vol.43, 1-7, 2009. (Summary) Let C be a given circle divided into n parts equally. Let C3 (n) be the number of incongruent triangles inscribed in C, where the vertices of the triangles are chosen from the ends of these parts. In this paper, we have shown a relation of this new defined number C3(n) and the partitions of n into at most three parts. (Tokushima University Institutional Repository) ● Metadata: 65207 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050001337463699584 (Tokushima University Institutional Repository: 65207, CiNii: 1050001337463699584)
21.	Shin-ichi Katayama : On some formulas for π/2, Journal of Mathematics, The University of Tokushima, Vol.42, 13-17, 2008. (Summary) In this paper, we have generalaized J.G. Goggins's formula involving inverse of Fibonacci numbers to general binary recurrences sequences. (Tokushima University Institutional Repository) ● Metadata: 62141 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050845762393825024 (Tokushima University Institutional Repository: 62141, CiNii: 1050845762393825024)
22.	Shin-ichi Katayama : Some infinite series of Fibonacci numbers, Journal of Mathematics, The University of Tokushima, Vol.42, 9-12, 2008. (Summary) In this paper, we have generalized J.G. Goggins's formula involving arctangent of inverse of Fibonacci numbers. (Tokushima University Institutional Repository) ● Metadata: 62140 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1050282812440403200 (Tokushima University Institutional Repository: 62140, CiNii: 1050282812440403200)
23.	Shin-ichi Katayama : On finite simple groups of cube order, Journal of Mathematics, The University of Tokushima, Vol.41, No.0, 25-31, 2007. (Summary) In this paper, we have modified the previous methods developed by M. Newman, D. Shanks and H.C. Williams on the order of finite symplectic groups, and shown that there is no symplectic group of cube order. (Tokushima University Institutional Repository) ● Metadata: 57945 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1570291227348187136 (Tokushima University Institutional Repository: 57945, CiNii: 1570291227348187136)
24.	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. (Summary) In this paper, we have determined the structure of integer solutions of the diophantine equation (x^2+1)(y^2+1)=(z^2+1)^2. (Tokushima University Institutional Repository) ● Metadata: 53568 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1572824501802327168 (Tokushima University Institutional Repository: 53568, CiNii: 1572824501802327168)
25.	Shin-ichi Katayama and Hiroharu Yasui : Maps R:Z/nZ → Z/n^2Z and some cryptographic applications, Journal of Mathematics, The University of Tokushima, Vol.39, 7-21, 2005. (Summary) In this paper, we have generalized the RSA signatures with redundancy functions develped in the first author's master thesis. We also investigated arithmetic properties of these redundancy functions and the RSA sgnatures with these redundancy functions. (Tokushima University Institutional Repository) ● Metadata: 49023 (Tokushima University Institutional Repository: 49023)
26.	Shin-ichi Katayama : A conjecture on fundamental units of real quadratic fields, Journal of Mathematics, The University of Tokushima, Vol.35, 9-15, 2002. (Summary) In our previous paper Proc.Japan Acad. 75A(2000), we have shown, under the ABC conjecture, the explicit units corresponding to Fibonacci numbers are fundamental units except for finitely many indices. We have actually verified all the explicit units for the indices leth than 701 are fundamental units. (Tokushima University Institutional Repository) ● Metadata: 141 (Tokushima University Institutional Repository: 141)
27.	Shin-ichi Katayama : Zeta functions of finite groups and arithmetically equivalent fields, Journal of Mathematics, The University of Tokushima, Vol.35, 1-8, 2002. (Summary) We consider all the finite group of order n(G,G',...) to be the subgroups of S_{n} by the regular representation. We also fix a S_{n} extension of Q and K,K',... to be the intermediate fields corresponding to G,G',... by Galois correspondence. Then zeta_S(G,s)=zeta_S(G',s) if and only if (S_{n},G) and (S_{n},G') are Gassmann equivalent and zeta_{K}(s)=zeta_{K'}(s) if and only if G and G' are isomorphic. (Tokushima University Institutional Repository) ● Metadata: 140 (Tokushima University Institutional Repository: 140)
28.	Shin-ichi Katayama : Several methods for solving simultaneous Fermat-Pell equations, Journal of Mathematics, Tokushima University, Vol.33, 1-14, 2000. (Summary) We determined the integer solutions of certain simultaneous Pell equations whichwere treated in Manuscripta Math. 105(2001) by the several different methods including Bakaer's method. (Tokushima University Institutional Repository) ● Metadata: 133 (Tokushima University Institutional Repository: 133)
29.	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. (Summary) We generalized the results on the distribution and the estimates of class numbers of the real qudratic fields of Chowla type by Montogomery and Weinberger to the real quadratic fields of extended Richaud-Degert type. (Tokushima University Institutional Repository) ● Metadata: 120 (Tokushima University Institutional Repository: 120)
30.	Shin-ichi Katayama : Unit groups of some quartic fields, Journal of Mathematics, Tokushima University, Vol.28, 1-4, 1995. (Summary) We treated new family of bicyclic biquadratic fields different from the quartic fields treated in Math. Jap. 42(1994), which were also related to Fibonacci numbers. We determined so called Hasse index and the structure of unit groups and obtained the class number formulas explicitly. (Tokushima University Institutional Repository) ● Metadata: 103 (Tokushima University Institutional Repository: 103)
31.	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. (Summary) We determined all the bicyclic biquadratic fields with class number one and two(which are the composite fields of real qudratic fields of Yokoi-Chowla type), under the generalized Riemann Hypothesis. (Tokushima University Institutional Repository) ● Metadata: 95 (Tokushima University Institutional Repository: 95)
32.	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. (Summary) We have shown the integer solutions of the diophantine equation z^2=(x^2-1)(y^2-1)+a are the orbit of certain infinite group acting on the representatives in certain(easily determined) fundamental domain. We also calculated all the representatives for the cases -144 eq a eq 148. (Tokushima University Institutional Repository) ● Metadata: 89 (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. (Summary) 小出長十郎の『砲術玉道真法』の付録とし小出由岐太(長十郎の養子)が著した「求玉道間数法」が，円理(円の面積，周長の積分)の応用として級数の項別積分を用いて放物線の長さの計算を行っていることと，現代の数学の視点で見ても興味深いことを紹介した． (Keyword) 求玉道間数法 / 円理 / 項別積分
2.	Shin-ichi Katayama : 海野十三と虫食い算, 徳島新聞, May 2015. (Keyword) 数学パズル / 科学史
3.	Shin-ichi Katayama : 海野十三と暗号, 徳島新聞, May 2014. (Keyword) 暗号理論 / 科学史
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. (Keyword) 整数論 / 不定方程式 (Link to Search Site for Scientific Articles) ● CiNii @ National Institute of Informatics (CRID): 1521699228471346560 (CiNii: 1521699228471346560)

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. (Summary) ある有限単純群の位数が，ベキ乗になることの必要十分条件は，その単純群がSymplectic Groupで，かつ平方数であるようなNeuman, Shanks, H.C. Williams によって見つけられたfamily以外にないことを示した．この事実からいわゆるNSW素数とベキ乗の位数を持つ有限単純群が1対1対応することを示した．
2.	Shin-ichi Katayama : On finite simple groups of square order, 1078th American Mathematical Society Sectional Meeting, Honolulu, Mar. 2012. (Summary) Shanks 等によって30年ほど前に一部の場合だけ決定された平方位数をもつ有限単純群の予想についてその予想を肯定的に決定した. すなわち彼らが見つけたfamily以外に平方数を持つ有限単純群は存在しないことを決定した．決定にはShanks等が用いた手法だけではなく最近発展したBugeaudらの不定方程式に関する手法, 著者によるbinary recurrence sequences の平方数の決定に関する工夫等の新しい手法が用いられた.
3.	Shin-ichi Katayama : On a family of simultaneous Pell equations, The seventh Japan-Korea Number Theory Symposium, Kujyu, Oct. 2004. (Summary) We have shown for a family of simultaneous Pell equations have at most one positive integer solution. We also show the only positive integer solution corresponds to the fundamtal units of real subquadratic fields of certain family of real bicyclic quadratic fields.
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. (Summary) Applying Kishi's construction of cyclic p-1 fields of class number divisible by p to Katayama's special bicyclic biquadratic fields, we have constructed infinitely many cyclic extensions of degree p-1 withthe p-ranks of the class groups st least 2.
5.	Shin-ichi Katayama : Zeta functions of finite groups and arithmetically equivalent fields, Seaway Number Theory Conference, Otawa (Canada), May 2001. (Summary) We gave a talk on the part of the results in J.Math.Univ. Tokushima 35(2002) and Ad.Stud. Contemp.Math. 4(2002).
6.	Shin-ichi Katayama : The abc conjecture, fundamental units and the simultaneous Pell equations, The fifth Japan-Korea Number Theory Symposium, Sendai, Nov. 1999. (Summary) We improved M.A.Bennett's upper estimate of the positive integer solutions of certain simultaneius Pell equations(3) to be 1 under the ABC conjecture using the structure of unit groups of certain family of real quadratic fields.
7.	Shin-ichi Katayama : Units of certain quartic fields and integer points on an elliptic curve, Workshop on Number Theory, Tokyo, Jul. 1999. (Summary) We talked on the results in Proc.Japan Acad. 75A(1999) as a point of view such that we actually determined integer points of certain elliptic curves using the Hasse principle on the square of rational integers.
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. (Summary) we gave a talk that we can generalize results of Math.Jap. 42(1995) to certain families of biquadratic fields.
9.	Shin-ichi Katayama : On composite real quadratic fields with class number one, 2nd Japan-Korea Symposium on Number Theory, Saga, Dec. 1996. (Summary) We gave a talk on the generalization of our results in J.Math.Tokushima Univ. 26(1993) to the composite real quadratic fields and some relation to capitulation problems in intermediate fields of Hilbert class fields.
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. (Summary) We gave a survey talk on th results of J.Math.Tokushima Univ.31(1998) and Adv.Stud.Contemp.Math.1(1999).

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. (Summary) Corner the Queen Gameは, Wythoffの石取りゲームと同値なゲームである．そのクイーンをナイトで取り換えた場合のゲームの勝敗，グランディー数を決定した． (Keyword) 石取りゲーム
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. (Summary) 五角数でありかつ平方数でありさらに三角数であるような数は，1しかないことについて2次の線形漸化式中の平方数の決定問題として解く方法と楕円胸腺上の整数点の決定問題として解く方法の2通りで決定した． (Keyword) ペル方程式 / 楕円曲線
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. (Summary) 江戸時代の和算のゲームである「碁石ひろい」の歴史および現在の数学の研究対象として，計算複雑性に関するNP完全性についてと取り尽くし可能ないくつかの無限系列の存在とその総取り尽くし可能数の評価式についての筆者の最近の成果を報告した．
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. (Summary) 和算における「碁石ひろい」の階段積みと呼ばれるある無限系列は，取り尽くし可能であることと，取り尽くし可能な特殊な手法を数えるとフィボナッチ数が現れることを示した． (Keyword) 石取りゲーム
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. (Summary) ここでは，除去可能な石の数を片方の山のみ有限パターンに制限し，他方の山の石を固定するとグランディー数が周期性を持つという場合に一般化できることを示した． (Keyword) 石取りゲーム
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. (Summary) Wytohoffの二山崩しのグランディー数は，帰納的に定義されるが，閉じた式で表す事は未だ未解決の問題である．ここでは，除去可能な石の数を有限に制限すると，片方の山の石を固定するとグランディー数が周期性を持つことを示し，いくつかの予想についても言及した． (Keyword) 石取りゲーム
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. (Summary) 海野十三の数学パズルへ②特異な嗜好として，彼の暗号小説中に現れる虫食い算とグリル暗号についてその仕組みを解説し紹介した．また佐野昌一名義で著した「虫食い算大会」の内容解説と完全虫食い算への系譜から現代に至るまでの彼の及ぼした影響と故郷徳島との関係についても紹介した．
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. (Summary) 海野十三の「荒唐無稽」への志向として，彼の暗号小説中に現れる虫食い算とグリル暗号についてその仕組みを解説し紹介した．また佐野昌一名義で著した「虫食い算大会」の内容解説と完全虫食い算への系譜から現代に至るまでの彼の及ぼした影響についても紹介した．
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. (Summary) 有限単純群 G の位数がべき乗になるための必要十分条件は，Gが4次の射影シンプレクティック群で，さらに基礎体がNSW素数位数の素体である時に限ると言う我々の最近の結果を紹介した．また現在知られているNSW素数のリストならびにNSW数が線形漸化式をみたすことを用いて，NSW数にLucasの素数判定法を適用する際に有用な性質を紹介した．
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. (Summary) フィボナッチの残している2つの書籍「Liber Abachi」(1202)，「De Practica Geometrie」(1220)の内容を簡単に紹介し，ラテン語の原文も参照しながらフィボナッチの行った円周率計算でのヨーロッパでの円周率計算の歴史における役割を説明した．
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. (Summary) 徳島出身の科学小説家である海野十三の暗号に関する小説について，特に「暗号の役割」で使われている暗号の特徴と日本の暗号小説における海野十三の特徴について解説した．
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. (Summary) フィボナッチの残した2つの書籍「Liber Abachi」(1202)，「De Practica Geometrie」(1220)から，フィボナッチの行った円周率計算がヨーロッパ社会に果たした役割を新たに指摘した．
13.	Shin-ichi Katayama : On finite simple groups of power order, Workshop in Number Theory in Saga 2011, Jan. 2011. (Summary) Shanks 等によって30年ほど前に一部の場合だけ決定された平方位数をもつ有限単純群の予想についてその予想を決定した. 決定にはShanks等が用いた手法だけではなく最近発展しているのBugeaudらの不定方程式に関する手法, 著者によるbinary recurrence sequences の平方数の決定に関する工夫等が用いられる.
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. (Summary) RDタイプの実2次体の類数問題に関する最近の結果の紹介と生成多項式から見た実2次体全体の中でのRDタイプの実2次体のある種の密度について報告した．
15.	Shin-ichi Katayama : Class number problem of quadratic fields, Galois theory and related problems Kanazawa 2009, Dec. 2009. (Summary) We gave a survey on the class number one problems for quadratic fields.
16.	Shin-ichi Katayama : Two topics on Gauss's half system, Number Theory in Saga 2009, Jan. 2009. (Summary) We gave a talk on two topics on Gauss's half system. The first topic is on a new visual proof of the reciprocity laws on Jacobi symbols. The second topic is a new relation between D. H. Lehmers problem and certain redundancy functions. We note that the first topics based on the master thesis of Mr. Arata Kuroki of Tokushima University.
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. (Summary) We gave a talk on certain type of distributuions of real quadratic fields of Richaud-Degert type in the set of real quadratic fields. We investigated the density of RD-type real quadratic fields via this special distribution.
18.	Shin-ichi Katayama and Hiroharu Yasui : RSA Signatures and Redundancy, 10th Algorithm on Number Theory and Its Applications, May 2003. (Summary) We can protect the multiplicative attack to the RSA signatures using the RSA sgnatures with redundancy of plain text of the form xx. We can estimate the strength of the protection using the arithmetic functions such as Euler functions and divisor functions.
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. (Summary) We gave a talk that the unit of real quadratic fields related to Fibonacci numbers obtained in Math.Jap. 42 (1995) and Proc.Japan Acad. 75A(1999) are the fundamental units with finitely many exceptions.
20.	Shin-ichi Katayama : Group rings and the norm groups of cyclic extensions, Nagoya Number Theory Seminar Spring Colloquim, Apr. 1993. (Summary) We gave a talk on the results of Proc. Japan Acad. 67A(1993).

Report:

1.	Shin-ichi Katayama : Class number problem of quadratic fields, Proceedings of the 8th Workshop on Number Theory, 81-96, Sep. 2010. (Summary) We gave a survey on the class number one problems for quadratic fields.
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. (Summary) We have shown for a family of simultaneous Pell equations have at most one positive integer solution. We also show the only positive integer solution corresponds to the fundamtal units of real subquadratic fields of certain family of real bicyclic quadratic fields.
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. (Summary) We gave a talk on the results of Proc.Japan. Acad. 75A(1999).
5.	Shin-ichi Katayama : On the class numbers od algebraic tori, Proceedings of the 33th symposium on algebra, 1-12, Fukui, Jul. 1987. (Summary) We gave a talk on the results obtained in Nagoya Math. Journal (1989).
6.	Shin-ichi Katayama : On the Galois cohomology groups of C_K / D_K, RIMS Kokyuroku, Vol.440, 2-10, Kyoto, Jun. 1981. (Summary) We gave a talk on the results of Japanese Journal of Math. 8(407-415).

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 )
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Professor Emeritus : Katayama, Shin-ichi

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