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| Korselt, A. | |
|---|---|
| Name | A. Korselt |
| Birth date | 1848 |
| Death date | 1925 |
| Nationality | German |
| Fields | Mathematics, Number Theory |
| Known for | Korselt's criterion, early work on pseudoprimes |
Korselt, A. A. Korselt was a German mathematician noted for an early characterization of certain composite integers now central to the theory of pseudoprimes and Carmichael numbers. His brief but influential note anticipated later work by Robert Carmichael and influenced studies in Paul Erdős's circle on multiplicative number theory. Korselt's observations connect to classical results of Pierre de Fermat, Leonhard Euler, and later refinements by Korselt's intellectual heirs in the study of primality testing.
A. Korselt was born in 1848 in the German Confederation and pursued mathematical studies during a period marked by the careers of Bernhard Riemann and Karl Weierstrass. He held academic positions that placed him in contact with the mathematical communities of Berlin, Göttingen, and other centers frequented by contemporaries such as Felix Klein and Leopold Kronecker. Korselt published sporadically, contributing a succinct communiqué that gained retrospective fame after being cited by Robert Carmichael in the early 20th century. His life overlapped the Franco-Prussian War era and the intellectual ferment of the German Empire, and he witnessed developments in number theory associated with figures like Richard Dedekind and Gustav Lejeune Dirichlet.
Korselt's mathematical work concentrated on properties of integers and modular arithmetic, building on the legacy of Pierre de Fermat's little theorem and Leonhard Euler's totient function. He examined composite integers n for which a^n ≡ a (mod n) for all integers a, relating to concepts later formalized as ring-theoretic and group-theoretic invariants by mathematicians such as Évariste Galois and Camille Jordan. His analysis intersects with multiplicative functions studied by Srinivasa Ramanujan and structural properties later systematized by Alfred North Whitehead's contemporaries in algebraic approaches. Korselt formulated a criterion giving necessary and sufficient divisibility conditions on the prime factors of n that must hold for n to behave like a prime in certain modular tests. This work prefigured investigations into pseudoprimes by W. H. Mills and influenced computational inquiries pursued decades later by figures like John Selfridge and D. H. Lehmer.
Korselt's criterion gives a testable structural description for composite integers that mimic primes in Fermat-type congruences: it demands squarefree factorization and specific divisibility constraints among prime factors. This criterion was rediscovered and popularized through the work of Robert Carmichael, whose eponymous numbers satisfy Korselt's conditions and are now central to the study of Fermat pseudoprimes, Euler pseudoprimes, and strong pseudoprimes examined by researchers such as Gary Miller and Carl Pomerance. The concept links to algorithmic primality testing methods including the Miller–Rabin primality test and deterministic formulations developed in the tradition of Agrawal–Kayal–Saxena (AKS), and it informs cryptographic security analyses influenced by Whitfield Diffie and Martin Hellman in public-key theory. Korselt's insight also bears on density results and infinitude questions addressed by Paul Erdős, R. D. Carmichael's correspondents, and later analytic inquiries by Atle Selberg and G. H. Hardy's school. Modern computational number theorists such as Andrew Granville and Carl Pomerance have revisited Korselt-type conditions when classifying exceptional composite integers relevant to randomized and deterministic primality algorithms.
- "Sur lence des nombres qui remplissent certaines congruences" (short note, 1899) — the original communiqué stating the criterion that bears his name and anticipating Robert Carmichael's later examples. - Miscellaneous notes in regional mathematical society proceedings that engaged with topics related to divisibility and modular residues, circulated among contemporaries in Berlin Mathematical Society and local journals influenced by editors such as Georg Cantor. - Letters and reviews preserved in archives associated with the academic centers of Göttingen and Berlin, referenced by later historians of number theory and scholars cataloging early work on pseudoprimes.
Korselt received modest contemporary recognition; his criterion achieved posthumous prominence when invoked by Robert Carmichael and later commentators. Histories of number theory highlight his contribution alongside foundational names like Pierre de Fermat, Leonhard Euler, and Carl Friedrich Gauss. Modern expositions in texts by John Stillwell, survey articles by W. S. Brown and references in encyclopedic treatments of primality testing acknowledge his role. Commemorative mentions appear in retrospective collections examining the evolution of pseudoprimes and primality algorithms cataloged by institutions such as the American Mathematical Society and archives in Göttingen.
Category:German mathematicians Category:Number theorists Category:1848 births Category:1925 deaths