{"product_id":"9780792363798","title":"Mathematics and Its Applications","description":"\u003ch1\u003eMathematics and Its Applications\u003c\/h1\u003e \u003ch2\u003eUrbanowicz, J.; Williams, Kenneth S.\u003c\/h2\u003e \u003cp\u003eIn [Hardy and Williams, 1986] the authors exploited a very simple idea to obtain a linear congruence involving class numbers of imaginary quadratic fields modulo a certain power of 2. Their congruence provided a unified setting for many congruences proved previously by other authors using various means. The Hardy-Williams idea was as follows. Let d be the discriminant of a quadratic field. Suppose that d is odd and let d = PIP2· . . Pn be its unique decomposition into prime discriminants. Then, for any positive integer k coprime with d, the congruence holds trivially as each Legendre-Jacobi-Kronecker symbol (~) has the value + 1 or -1. Expanding this product gives ~ eld e:=l (mod4) where e runs through the positive and negative divisors of d and v (e) denotes the number of distinct prime factors of e. Summing this congruence for o \u0026lt; k \u0026lt; Idl\/8, gcd(k, d) = 1, gives ~ (-It(e) ~ (~) =:O(mod2n). eld o\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 2000-06-30\u003c\/p\u003e \u003cp\u003eFormat: Hardcover\u003c\/p\u003e \u003cp\u003eISBN-13: 9780792363798\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-94-015-9542-1\u003c\/p\u003e \u003cp\u003eDimensions: 234cm x156cm\u003c\/p\u003e \u003cp\u003ePages: 256\u003c\/p\u003e ","brand":"Springer Netherlands","offers":[{"title":"Default Title","offer_id":46539697815692,"sku":"9780792363798","price":49.49,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9780792363798.jpg?v=1775679108","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9780792363798","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}