Matches in SemOpenAlex for { <https://semopenalex.org/work/W1984458434> ?p ?o ?g. }
Showing items 1 to 69 of
69
with 100 items per page.
- W1984458434 abstract "There exist infinitely many positive integers n such that 2~2 = 1 (mod n). In the monograph (5) I proposed the following problem (problem 18, p. 138): Let a, k > 1 be fixed positive integers. Do there exist infinitely many composite n such that «| a*- 1? Put a = 2, k = in the above problem. Since by Fermat's theorem 2p~l = 1 (mod p) for odd primes p, if = 1 (mod n) and n > 2, n must be composite. R. Matuszewski and P. Rudnicki (with the aid of the computer K-202 in Warsaw) checked that below 4208 such integers do not exist. The following theorem holds: congruence 2m = 1 (mod2m - 1) it follows that (2m)V-V/m = 1 (mod2m - 1), 2213 s 1 (mod2m - 1) and 2~2 = 1 (modn) for n = 2m - 1. Thus 22 = 1 (mod n) for n = 2» - 1, where «0 = 4700063497. Suppose now that 22 = 1 (mod «), and n > 8. Let/? be a primitive factor of the number 2~2 - 1 (a prime factor of 2 - 1 is said to be primitive if it does not divide any of the numbers 2m - 1 for m = 1,2,...,n — 1. By a theorem of K. Zsigmondy (7) such a prime factor exists for any n > 6 and is of the form nt + 1). Now we shall show that nx = np is also a solution of the congruence 21_2 = 1 (mod «,). We have p = 2(n - 2)k + 1, where k is a positive integer and p > 2« - 3 > n and (/?,«)= 1. Thus" @default.
- W1984458434 created "2016-06-24" @default.
- W1984458434 creator A5051047716 @default.
- W1984458434 date "1984-07-01" @default.
- W1984458434 modified "2023-09-27" @default.
- W1984458434 title "On the Congruence 2 n - 2 ≡1 (mod n)" @default.
- W1984458434 cites W1489006728 @default.
- W1984458434 cites W1555475515 @default.
- W1984458434 cites W1991451613 @default.
- W1984458434 cites W2319459379 @default.
- W1984458434 doi "https://doi.org/10.2307/2007412" @default.
- W1984458434 hasPublicationYear "1984" @default.
- W1984458434 type Work @default.
- W1984458434 sameAs 1984458434 @default.
- W1984458434 citedByCount "6" @default.
- W1984458434 countsByYear W19844584342018 @default.
- W1984458434 crossrefType "journal-article" @default.
- W1984458434 hasAuthorship W1984458434A5051047716 @default.
- W1984458434 hasConcept C114614502 @default.
- W1984458434 hasConcept C118615104 @default.
- W1984458434 hasConcept C132074034 @default.
- W1984458434 hasConcept C184992742 @default.
- W1984458434 hasConcept C199360897 @default.
- W1984458434 hasConcept C2524010 @default.
- W1984458434 hasConcept C29231244 @default.
- W1984458434 hasConcept C30860621 @default.
- W1984458434 hasConcept C33923547 @default.
- W1984458434 hasConcept C41008148 @default.
- W1984458434 hasConcept C87670640 @default.
- W1984458434 hasConcept C97137487 @default.
- W1984458434 hasConceptScore W1984458434C114614502 @default.
- W1984458434 hasConceptScore W1984458434C118615104 @default.
- W1984458434 hasConceptScore W1984458434C132074034 @default.
- W1984458434 hasConceptScore W1984458434C184992742 @default.
- W1984458434 hasConceptScore W1984458434C199360897 @default.
- W1984458434 hasConceptScore W1984458434C2524010 @default.
- W1984458434 hasConceptScore W1984458434C29231244 @default.
- W1984458434 hasConceptScore W1984458434C30860621 @default.
- W1984458434 hasConceptScore W1984458434C33923547 @default.
- W1984458434 hasConceptScore W1984458434C41008148 @default.
- W1984458434 hasConceptScore W1984458434C87670640 @default.
- W1984458434 hasConceptScore W1984458434C97137487 @default.
- W1984458434 hasLocation W19844584341 @default.
- W1984458434 hasOpenAccess W1984458434 @default.
- W1984458434 hasPrimaryLocation W19844584341 @default.
- W1984458434 hasRelatedWork W1486300046 @default.
- W1984458434 hasRelatedWork W1559932750 @default.
- W1984458434 hasRelatedWork W2006551572 @default.
- W1984458434 hasRelatedWork W2023898236 @default.
- W1984458434 hasRelatedWork W2028146943 @default.
- W1984458434 hasRelatedWork W2036092007 @default.
- W1984458434 hasRelatedWork W2079526217 @default.
- W1984458434 hasRelatedWork W2082598440 @default.
- W1984458434 hasRelatedWork W2123572819 @default.
- W1984458434 hasRelatedWork W2325751678 @default.
- W1984458434 hasRelatedWork W2325899845 @default.
- W1984458434 hasRelatedWork W2327638732 @default.
- W1984458434 hasRelatedWork W2328667888 @default.
- W1984458434 hasRelatedWork W2541959419 @default.
- W1984458434 hasRelatedWork W2953095682 @default.
- W1984458434 hasRelatedWork W2953165836 @default.
- W1984458434 hasRelatedWork W2962767533 @default.
- W1984458434 hasRelatedWork W3021705746 @default.
- W1984458434 hasRelatedWork W49606806 @default.
- W1984458434 hasRelatedWork W2183649670 @default.
- W1984458434 isParatext "false" @default.
- W1984458434 isRetracted "false" @default.
- W1984458434 magId "1984458434" @default.
- W1984458434 workType "article" @default.