LEADER 03347nam 22004695 450 001 9910999676403321 005 20250427130146.0 010 $a981-9625-99-8 024 7 $a10.1007/978-981-96-2599-4 035 $a(CKB)38641482300041 035 $a(DE-He213)978-981-96-2599-4 035 $a(MiAaPQ)EBC32029804 035 $a(Au-PeEL)EBL32029804 035 $a(EXLCZ)9938641482300041 100 $a20250427d2025 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aPerfect Powers?An Ode to Erd?s /$fby Saradha Natarajan 205 $a1st ed. 2025. 210 1$aSingapore :$cSpringer Nature Singapore :$cImprint: Springer,$d2025. 215 $a1 online resource (XIV, 184 p. 3 illus.) 225 1 $aInfosys Science Foundation Series in Mathematical Sciences,$x2364-4044 311 08$a981-9625-98-X 327 $aChapter 1 Preliminaries: A Tool Kit -- Chapter 2 Basic ideas of Erdös -- Chapter 3 Theorem of Sylvester. 330 $aThe book explores and investigates a long-standing mathematical question whether a product of two or more positive integers in an arithmetic progression can be a square or a higher power. It investigates, more broadly, if a product of two or more positive integers in an arithmetic progression can be a square or a higher power. This seemingly simple question encompasses a wealth of mathematical theory that has intrigued mathematicians for centuries. Notably, Fermat stated that four squares cannot be in arithmetic progression. Euler expanded on this by proving that the product of four terms in an arithmetic progression cannot be a square. In 1724, Goldbach demonstrated that the product of three consecutive positive integers is never square, and Oblath extended this result in 1933 to five consecutive positive integers. The book addresses a conjecture of Erd?s involving the corresponding exponential Diophantine equation and discusses various number theory methods used to approach a partial solution to this conjecture. This book discusses diverse ideas and techniques developed to tackle this intriguing problem. It begins with a discussion of a 1939 result by Erd?s and Rigge, who independently proved that the product of two or more consecutive positive integers is never a square. Despite extensive efforts by numerous mathematicians and the application of advanced techniques, Erd?s' conjecture remains unsolved. This book compiles many methods and results, providing readers with a comprehensive resource to inspire future research and potential solutions. Beyond presenting proofs of significant theorems, the book illustrates the methodologies and their limitations, offering a deep understanding of the complexities involved in this mathematical challenge. 410 0$aInfosys Science Foundation Series in Mathematical Sciences,$x2364-4044 606 $aNumber theory 606 $aNumber Theory 615 0$aNumber theory. 615 14$aNumber Theory. 676 $a512.7 700 $aNatarajan$b Saradha$4aut$4http://id.loc.gov/vocabulary/relators/aut$01017961 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910999676403321 996 $aPerfect Powers?An Ode to Erd?s$94374769 997 $aUNINA