number theorist names:G

1. István Gaál
   • Diophantine Equations and Power Integral Bases (Theory and Algorithms), István Gaál, Birkhaüser, 2019 (new, substantially extended second edition)
   • Diophantine equations and power integral bases, István Gaál, Birkhaüser, 2002
2. Mikhail Rashidovich Gabdullin
3. Wojciech Gajda
4. Stevan Gajović
5. Aurélien Galateau
6. Steven Galbraith
   • Mathematics of Public Key Cryptography, Steven Galbraith, Cambridge University Press, 2012
   • Equations For Modular Curves, PhD Thesis, Steven Galbraith, Oxford University, 1997
7. Luis Gallardo gallardo@math.cnrs.fr Luis.Gallardo@univ-brest.fr
8. Homero R. Gallegos-Ruiz
9. Yves Gallot
   • Proth.exe (Yves Gallot)
   • Generalized Fermat Primes Search
10. Will Galway
    • A Higher-Dimensional Sieve Method With Procedures for Computing Sieve Functions, Harold Diamond, Heini Halberstam, Will Galway, Cambridge Tracts in Mathematics 177 2008
11. Alexander Gamburd
12. Adam Gamzon
13. Wee Teck Gan
    • The Langlands Program: The Set (Volumes I, II, and III), Ed. Pierre-Henri Chaudouard, Wee Teck Gan, Tasho Kaletha, Yiannis Sakellaridis, AMS, 2026
    • Automorphic Forms Beyond GL₂: Lectures from the 2022 Arizona Winter School, Ellen Elizabeth Eischen, Wee Teck Gan, Aaron Pollack, Zhiwei Yun, Ed. Hang Xue, AMS Mathematical Surveys and Monographs Vol. 279, 2024
    • Eisenstein Series and Applications, Wee Teck Gan, Stephen S. Kudla, Yuri Tschinkel (Eds.) Progress in Mathematics 258, Birkhäuser 2007
14. Herbert Gangl
    • Table of tame and wild kernels of quadratic imaginary number fields of discriminants > -5000 (conjectural values), Jerzy Browkin and Herbert Gangl
15. Abhik Ganguli
16. Satadal Ganguly
17. Shuhong Gao
18. Ziyang Gao
    • Vesselin Dimitrov and Ziyang Gao win the 2022 David Goss Prize
19. Elisa Lorenzo García
20. Stephan Ramon Garcia
21. Natalia Garcia-Fritz
    • Curves of low genus and applications to Diophantine problems, PhD thesis, Natalia Cristina Garcia Fritz, Queen's University, Kingston, Canada, 2015
22. Lynn Garner
23. Paul Garrett
    • Holomorphic Hilbert Modular Forms, Paul Garrett, Wadsworth-Brooks-Cole, 1990, Review, Michael Harris, Bull. Amer. Math. Soc. 25 (1991), 184-195, Errata to the review
    • Modular Forms, T. Miyake, Springer 1989 (no longer listed by Springer), Review, Paul B. Garrett, Bull. Amer. Math. Soc. 24 (1991), 234-245
    • Vignettes on Automorphic and Modular Forms, Representation , L-functions and Number Theory
    • Bibliography for Automorphic and Modular Forms, L-functions and Representation Theory
24. Thomas Garrity
25. Jérôme Gärtner
    • Points de Darmon et variétés de Shimura, PhD thesis, Jérôme Gärtner, Paris 2011
26. Derek Garton
27. Ramunas Garunkstis
    • The Lerch zeta-function, A. Laurincikas, R. Garunkstis, Springer 2003
28. Frank Garvan
    • Partitions, q-Series, and Modular Forms, Ed. Krishnaswami Alladi, Frank Garvan, Series: Developments in Mathematics, Vol. 23, Springer 2012
29. Carlo Gasbarri
    • Arithmetic Geometry, Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10-15, 2007, Jean-Louis Colliot-Thélène, Peter Swinnerton-Dyer, Paul Vojta, Pietro Corvaja, Carlo Gasbarri (Eds.), Lecture Notes in Mathematics, Vol. 2009, 2010
30. Joachim von zur Gathen
    • CryptoSchool, Joachim von zur Gathen, Springer 2015
    • Modern Computer Algebra, J. von zur Gathen, J. Gerhard, CUP 1998
31. Éric Gaudron
32. Pierrick Gaudry
    • Explicit Methods in Number Theory Rational Points and Diophantine Equations, Karim Belabas, Hendrik W. Lenstra Jr., Pierrick Gaudry, Michael Stoll, Mark Watkins, William McCallum & Bjorn Poonen, Frits Beukers, Samir Siksek, Panoramas et synthèses 36 (2012)
33. Bruno Gauthier
    • HYPERG V 1.0, Maple package devoted to symbolic computation on hypergeometric series (Bruno Gauthier)
    • GUESS, Maple package to guess closed form for a sequence of numbers (Bruno Gauthier)
34. Sébastien Gauthier
35. Quentin Gazda
36. Fan Ge
37. Max Gebhardt
38. Toby Gee
39. Gerard van der Geer
    • Modular Forms on Schiermonnikoog, Ed. Bas Edixhoven, Gerard van der Geer, Ben Moonen, CUP 2008
    • Number Fields and Function Fields - Two Parallel Worlds, Ed. Gerard van der Geer, Ben Moonen, Progress in Mathematics, 239, Birkhäuser 2005
    • Mathematics Unlimited - 2001 and Beyond, B. Engquist, W. Schmid, (Ed), Springer 2000 (contains articles by Darmon, Faltings, Koblitz, Van der Geer)
    • Hilbert Modular Forms, E. Freitag, Springer 1990, Review, Gerard van der Geer, Bull. Amer. Math. Soc. 25 (1991), 441-445
    • Hilbert Modular Surfaces, G. van der Geer, Ergebnisse der Mathematik 16, Springer 1988, Review, Gerd Faltings, Bull. Amer. Math. Soc. 20 (1989), 247-251
    • Introduction to Coding Theory and Algebraic Geometry, Jacobus H. van Lint, Gerard van der Geer, Oberwolfach Seminars, 12, Birkhäuser 1988
40. Lennart Gehrmann
41. Ernst-Ulrich Gekeler
42. Stephen Gelbart
    • On Certain $L$-Functions, Ed. James Arthur, James Cogdell, Steve Gelbart, David Goldberg, Dinikar Ramakrishnan, Jiu-Kang Yu, 13, Clay Mathematics Proceedings, 2011
    • Review by Stephen Gelbart of Advanced Analytic Number Theory: L-Functions, Carlos Moreno
    • Riemann's zeta function and beyond, Stephen Gelbart, Stephen Miller, Bull. AMS 41 (2004), 59-112
    • An Introduction to the Langlands Program, Ed. Joseph Bernstein, Stephen Gelbart, Birkhaüser 2003
43. Tyler Genao
44. David Geraghty
45. Jürgen Gerhard
    • Modern Computer Algebra, J. von zur Gathen, J. Gerhard, CUP 1998
46. Alfred Geroldinger
    • Combinatorial Number Theory and Additive Group Theory, Alfred Geroldinger and Imre Z. Ruzsa, Birkhäuser 2009
    • Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Alfred Geroldinger and Franz Halter-Koch, Chapman & Hall/CRC 2006
47. Maxim Gerspach
48. Ellen Gethner
49. Jayce Getz
50. Betül Gezer
51. Wissam Ghantous
52. Eknath Ghate
    • Eknath Ghate Awarded 2013 Shanti Swarup Bhatnagar Prize
    • The arithmetic and geometry of Salem numbers, E. Ghate and E. Hironaka, Bulletin AMS 38 (2001), 293-314
    • Critical values of the twisted tensor L-function in the imaginary quadratic case, PhD Thesis, Eknath Ghate, UCLA 1996
53. Adela Gherga
54. Dragos Ghioca
55. Alex Ghitza
    • The abc conjecture, as easy as 1, 2, 3 ⋯ or not, Alex Ghitza, The Conversation, 26 November 2012
    • Siegel modular forms (mod p) and algebraic modular forms, PhD thesis, Alexandru Ghitza, MIT 2003
56. Amit Ghosh
    • Topics in prime number theory, PhD. thesis, Amit Ghosh, Nottingham 1981
57. Anish Ghosh
58. Arkabrata Ghosh
59. Sohan Ghosh
60. Badih Ghusayni
    • Number Theory from an analytic point of view (Théorie de Nombre d'un point de vue analytique), Badih Ghusayni
61. Average Frobenius Distributions for Elliptic Curves: Extremal Primes and Koblitz's Conjecture, PhD thesis, Luke M. Giberson, Clemson University, 2017
62. Alexandru Gica
63. Mark Giesbrecht
64. R. J. Hans Gill
65. Roland Gillard
66. Sally Gilles
67. Timothy Gillespie
    • Superposition of zeros of automorphic L-functions and functoriality, PhD thesis, Timothy Gillespie, University of Iowa, 2011
68. Henri Gillet
69. Jean Gillibert
    • Invariants de classes pour les variétés abéliennes à réduction semi-stable, PhD thesis, Jean Gillibert, Université de Caen, 2004
70. Martine Girard
    • Points de Weierstrass et jacobiennes de courbes algébriques de genre 3, PhD thesis, Martine Girard, Paris VII 2000
71. Sumit Giri
72. Kurt Girstmair
73. Bernhard Gittenberger
74. Jacob Glas
75. Darren Glass
    • Orthogonal Epsilon Constants For Tame Actions of Finite Groups on Surfaces, PhD thesis, Darren Glass, University of Pennsylvania 2002
76. Amy Glen
77. Bart Goddard
78. Hemar Godinho
    • Teoria dos números, S. Shokranian, M. Soares, H. Godinho, Editora Universidade de Brasilia (1994)
79. Eva Goedhart
80. Shivani Goel
81. Edray Goins
82. Robert Gold
83. Dorian Goldfeld
    • Automorphic Forms and L-Functions for the Group GL(n,R), Dorian Goldfeld, Cambridge Studies in Advanced Mathematics 99, CUP 2015
    • Multiple Dirichlet Series, L-functions and Automorphic Forms, Ed. Daniel Bump, Solomon Friedberg, Dorian Goldfeld, Progress in Mathematics Vol. 300, Birkhäuser Basel, 2012
    • Number Theory, Analysis and Geometry, In Memory of Serge Lang, Springer 2012
    • Automorphic Representations and L-Functions for the General Linear Group, Vol. 1, Dorian Goldfeld and Joseph Hundley, Cambridge Studies in Advanced Mathematics 129, 2011
      • Review by Ramin Takloo-Bighash of Automorphic representations and L-functions for the general linear group, Volume I, by Dorian Goldfeld
    • Automorphic Representations and L-Functions for the General Linear Group, Vol. 2, Dorian Goldfeld and Joseph Hundley, Cambridge Studies in Advanced Mathematics 130, 2011
    • Collected Works of Hervé Jacquet, Ed. Dorian Goldfeld, AMS Collected Works , Volume 23, 2011
    • Photos from the 60th Birthday conference for Dorian Goldfeld, Columbia University New York, May 18-23, 2007 (C.J. Mozzochi)
    • Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, Ed. Solomon Friedberg, Daniel Bump, Dorian Goldfeld, Jeffrey Hoffstein, Proceedings of Symposia in Pure Mathematics 75, AMS, Dec 2006
    • Automorphic Forms and L-Functions for the Group GL(n,R), Dorian Goldfeld, CUP, 2006
    • Explicit Formulas for Regularized Products and Series, J.A. Jorgenson, S. Lang, D. Goldfeld, Lecture Notes in Mathematics 1593, Springer 1994, ISBN 3-540-56873-3
    • Dorian Goldfeld: The Gauss class number problem, Bull. Amer. Math. Soc. 11, July 1985
84. Leo Goldmakher
85. Luca Goldoni
    • Prime Numbers and Polynomials, PhD thesis, Luca Goldoni, University of Trento, 2010
86. Catherine Goldstein
    • The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae, (Ed.) Catherine Goldstein, Norbert Schappacher, Joachim Schwermer, Springer 2006
    • Le Théorème de Fermat (Karim Belabas and Catherine Goldstein)
87. Larry Joel Goldstein
    • Introduction to Number Theory, Larry Joel Goldstein, William Wells Adams, Prentice-Hall, Inc. Englewood Cliffs, New Jersey (1976)
    • Analytic Number Theory, Larry Joel Goldstein, Prentice-Hall, Englewood Cliffs, New Jersey (1971)
88. Susan Goldstine
89. Dan Goldston
    • Yitang Zhang and Daniel Goldston, János Pintz, Cem Yalçin Yildirim receive the 2014 AMS Frank Nelson Cole Prize in Number Theory
90. Shafi Goldwasser
91. Carlos Alexis Gómez
92. Yasuro Gon
93. Felipe Gonçalves
94. Alexander Goncharov
95. Danielle Gondard
96. Steven Gonek
    • Three lectures on the Riemann zeta function (Steve Gonek)
97. Ke Gong
98. Enrique González-Jiménez
    • New modular non-hyperelliptic genus 3 curves (Enrique González Jiménez)
    • Modular Hyperelliptic Curves, PhD thesis, E. González-Jiménez, Universitat Autonoma de Barcelona, 2002
    • New Modular Hyperelliptic Curves (E. González-Jiménez)
    • Genus 2 curves with modular jacobians (E. González-Jiménez)
99. Haydar Göral
100. Heidi Goodson
101. Dan Gordon
102. Eyal Goren
     • Lectures on Hilbert Modular Varieties and Modular Forms, E.Z. Goren, AMS, 2002
     • Number Theory, Ed. H. Kisilevsky, E. Goren, CRM Proceedings & Lecture Notes 36, AMS 2004
     • Tables of Dedekind zeta functions at negative integers (Eyal Goren)
103. Ofir Gorodetsky
     • The variance of sums of arithmetic functions, PhD thesis, Ofir Gorodetsky, Tel Aviv University 2021
104. Alexander Gorodnik
     • Dynamics and Analytic Number Theory, Ed. Dzmitry Badziahin, Alexander Gorodnik, Norbert Peyerimhoff, LMS Lecture Note Series 457, CUP 2016
105. Takeshi Goto
     • Largest prime factor of an odd perfect number (Takeshi Goto)
     • Harmonic numbers (Takeshi Goto)
106. Frank Gounelas
107. Fernando Gouvêa
     • Common Inessential Discriminant Divisors: Scenes from the Early History of Algebraic Number Theory, Fernando Q. Gouvêa, Jonathan Webster, AMS 2025
     • Review of The Quadratic Reciprocity Law, a collection of classical proofs, Oswald Baumgart, Edited and Translated by Franz Lemmermeyer, Birkhäuser 2015, by Fernando Gouvêa
     • MAA review by Fernando Gouvêa of Peter Roquette's book The Riemann Hypothesis in Characteristic p in Historical Perspective
     • Review of Dan Flath's book: Introduction to Number Theory, by Fernando Q. Gouvêa
     • Review of Roquette's Contributions to the History of Number Theory in the 20th Century by Fernando Gouvêa
     • Review by Fernando Q. Gouvêa of Goro Shimura's The map of my life
     • A Short Course in Automorphic Functions, Joseph Lehner, Dover, 1966 (MAA review by Fernando Q. Gouvêa)
     • Review of Cohomology of Number Fields, J. Neukirch, A. Schmidt, K. Wingberg (Reviewer: F.Q. Gouvêa) Bulletin AMS 39, 101-107, 2002
     • Advances in Number Theory, (ed. F. Gouvêa, N. Yui), Proceedings of the Third Conference of the Canadian Number Theory Association, 1991, Kingston, OUP 1993
     • Third edition of p-adic Numbers: An Introduction, Fernando Gouvêa, MAA Review by Allen Stenger
     • Arithmetic of Diagonal Hypersurfaces over Finite Fields, F. Gouvêa, N. Yui, LMS Lecture Notes 209, CUP, 1995
     • Arithmetic of p-adic Modular Forms, F. Gouvêa, Lecture Notes 1304, Springer 1988
108. Timothy Gowers
     • A recent experience with ChatGPT 5.5 Pro from Gowers's Weblog
     • Analytic Number Theory, Essays in Honour of Klaus Roth, Ed. William Chen, Timothy Gowers, Heini Halberstam, Wolfgang Schmidt, Robert Vaughan, CUP 2009
     • Vinogradov's three-primes theorem, notes by Timothy Gowers (dvi 54K)
109. Peter Grabner
     • Number Theory - Diophantine Problems, Uniform Distribution and Applications, Ed. Christian Elsholtz, Peter Grabner, Springer 2017
110. Peter Gräf
111. Andrew Graham
112. Sid Graham
     • Sid Graham's Review of Prime Obsession by John Derbyshire
     • Sid Graham's Review of The Riemann Hypothesis by Karl Sabbagh
     • Van der Corput's Method for Exponential Sums, S.W. Graham and G. Kolesnik, LMS Lecture Notes 126, CUP 1991
113. Louis Granboulan
114. Robert Granger
115. David Grant
116. Jon Grantham
     • Frobenius Pseudoprimes (Jon Grantham)
117. Andrew Granville
     • Analytic Number Theory Revealed: A First Guide to Prime Numbers, Andrew Granville, AMS 2025
     • The Man Who Loved Problems: Richard K. Guy, Andrew Granville and Carl Pomerance, AMS Notices, April 2022
     • Number Theory Revealed: A Masterclass, Andrew Granville, Miscellaneous Books, Vol. 127, AMS, 2019
     • Number Theory Revealed: An Introduction, Andrew Granville, AMS Textbook, 2019, Review by Nathan McNew
     • Primes in intervals of bounded length, Andrew Granville, Bull. Amer. Math. Soc. 52 (2015), 171-222
     • Anatomy of Integers, ed. Jean-Marie De Koninck, Andrew Granville, Florian Luca, CRM Proceedings & Lecture Notes, 46, August 2008
     • Equidistribution in Number Theory, An Introduction, Proceedings of the NATO Advanced Study Institute on Equidistribution in Number Theory, Montreal, Canada, 11-22 July 2005, ed. Andrew Granville, Zeev Rudnick, NATO Science Series II: Mathematics, Physics and Chemistry 237, Springer 2007
     • It is easy to determine whether a given integer is prime, Andrew Granville, Bull. Amer. Math. Soc. 42 (2005), 3-38
     • Review of Alf van der Poorten's Notes on Fermat's Last Theorem by Andrew Granville
     • Andrew Granville's Binomial Page
     • A Tribute to Mathematical Legend Paul Erdös, K. Alladi, P.D.T.A. Elliott, A. Granville, G. Tenenbaum, Vol 1, Developments in Mathematics, Kluwer 1998
     • Review of BBC's Horizon program, "Fermat's Last Theorem" (Andrew Granville) Notices of the AMS, January 1997
118. Georges Gras
     • Class Field Theory: From Theory to Practice, Georges Gras, corr. 2nd ed., Springer Monographs in Mathematics, Springer 2005, xiii+507 pages
119. Marie-Nicole Gras
120. Hester Graves
     • Problems in the Theory of Modular Forms, M. Ram Murty, Michael Dewar, Hester Graves, Springer 2016
     • Problems in the Theory of Modular Forms, M. Ram Murty, Michael Dewar, Hester Graves, Hindustan Book Agency, 2015
     • On Euclidean Ideal Classes, PhD thesis, Hester K. Graves, U. Michigan 2009
121. Neven Grbac
122. Gary Greaves
     • Cyclotomic Matrices over Quadratic Integer Rings, PhD thesis, Gary Greaves, University of London, 2012
123. Barry Green
124. Ben Green
     • Additive combinatorics reviewed by Ben Green, Bull. Amer. Math. Soc. 46 (2009), 489-497
     • Ben Green to receive the 2007 SASTRA Ramanujan prize
125. Holly Green
126. Ralph Greenberg
     • Lectures and videos by John Coates, David Loeffler and Sarah Zerbes, Romyar Sharifi, Christopher Skinner and Ralph Greenberg, Arizona Winter School 2018: Iwasawa Theory
     • Iwasawa Theory, Projective Modules, and Modular Representations, Ralph Greenberg, 211, Memoirs of the American Mathematical Society, 2011
     • Memories of Professor Iwasawa, (Ralph Greenberg)
     • 120 minutes at the house of Professor Kenkichi Iwasawa (Ralph Greenberg)
     • Arithmetic theory of elliptic curves, J. Coates, R. Greenberg, K. Ribet, K. Rubin, C. Viola (Ed), Springer Lecture Notes 1716, 1999
127. Aaron Greicius
128. Cornelius Greither
     • Multiplicative Galois Module Structure, Afred Weiss, Fields Institute Monographs 6 1996, Review, Cornelius Greither, Bull. Amer. Math. Soc. 34 (1997), 173-176
129. George Grekos
130. Loïc Grenié
131. Sigrid Grepstad
132. Michael Griffin
133. Richard Griffon
     • Curves over finite fields, Richard Griffon
     • Analogues du théorème de Brauer-Siegel pour quelques familles de courbes elliptiques, PhD thesis, Richard Griffon, Université Paris Diderot, 2016
134. Harald Grobner
135. Rob Gross
     • Summing It Up: From One Plus One to Modern Number Theory, Avner Ash & Robert Gross, Princeton University Press 2016, MAA Review by Mark Hunacek
     • Elliptic Tales: Curves, Counting, and Number Theory, Avner Ash & Robert Gross, Princeton University Press, 2012
     • Fearless Symmetry: Exposing the Hidden Patterns of Numbers, Avner Ash & Robert Gross, Princeton University Press 2008, MAA review
136. Samuel Gross
     • Irreducibility Criteria for Polynomials With Non-negative Integer Coefficients and the Prime Factorization of f(n) for f(x) in ℤ[x], PhD thesis, Samuel Gross, University of South Carolina, 2012
137. Giada Grossi
138. David Gruenewald
     • Explicit Algorithms for Humbert Surface, PhD thesis, David Gruenewald, University of Sydney, 2009
139. Helen Grundman
140. Aleksander Grytczuk
141. Maciej Grześkowiak
142. Jordi Guàrdia
     • +Ideals, A package for ideal arithmetic in number fields (Jordi Guàrdia, Jesús Montes & Enric Nart)
143. Walter Gubler
     • Heights in Diophantine Geometry, Enrico Bombieri, Walter Gubler, New Mathematical Monographs 4, CUP 2006
144. Pavel Guerzhoy
     • Lecture on continued fractions (Pavel Guerzhoy - pdf)
145. Aurore Guillevic
146. Pierre Guillot
     • A Gentle Course in Local Class Field Theory Local Number Fields, Brauer Groups, Galois Cohomology, Pierre Guillot, CUP, 2018
147. Xevi Guitart
148. Ahmet Güloğlu
149. Fabian Gundlach
150. Keiichi Gunji
151. Paul Gunnells
152. Li Guo
153. Victor J. W. Guo
154. Xuejun Guo
155. Shamita Dutta Gupta
156. Shyam Sunder Gupta
     • Exploring the Beauty of Fascinating Numbers, Shyam Sunder Gupta, Springer 2025
     • Creative Puzzles to Ignite Your Mind, Shyam Sunder Gupta, Springer 2023
     • Happy numbers Shyam Sunder Gupta
     • Fascinating Factorials Shyam Sunder Gupta
     • Amazing number 108 Shyam Sunder Gupta
     • Smith numbers (Shyam Sunder Gupta)
     • Fascinating Triangular Numbers (Shyam Sunder Gupta)
157. Lance Gurney
158. Damián Gvirtz-Chen
159. Katalin Gyarmati
160. Kálmán Győry
     • Effective Results and Methods for Diophantine Equations over Finitely Generated Domains, Jan-Hendrik Evertse, Kálmán Győry, CUP 2022
     • Discriminant Equations in Diophantine Number Theory, Jan-Hendrik Evertse, Kálmán Győry, New Mathematical Monographs 32, CUP 2016
     • Unit equations in Diophantine number theory, Jan-Hendrik Evertse, Kálmán Győry, Cambridge Studies in Advanced Mathematics 146, CUP 2016, MAA Review
     • Number Theory, Analysis, and Combinatorics, Proceedings of the Paul Turan Memorial Conference held August 22-26, 2011 in Budapes, Ed. J, Pintz, A. Biró, K. Győry, G. Harcos, M. Simonovits, J. Szabados, De Gruyter Proceedings in Mathematics, 2013
     • Unit Equations in Diophantine Number Theory, Jan-Hendrik Evertse, Kálmán Győry, CUP 2015, MAA Review of Unit Equations in Diophantine Number Theory
     • Number Theory and Its Applications, Ed. S. Kanemitsu, K. Győry, Developments in Mathematics 2, Kluwer 1999

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Last modified 7th July 2026