Kummer Congruences for Products of Numbers

✍ Scribed by Harlan Stevens

• Publisher: John Wiley and Sons
• Year: 1962
• Tongue: English
• Weight: 333 KB
• Volume: 24
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19620240402

✦ Synopsis

Let R, represent the set of all rational numbers which are integral (mod m ) . We recall that a/b is integral (mod m) if m and b are relatively prime. If {a,L} is a sequence of numbers in R,, where p is a fixed rational prime, it is customary to say that a, satisfies KUMMER'S congruence when

for all n 2 r 2 1, where 2 is also in R, . For example (1.1) holds when p > 2, 1 = 1 and a, = En, the EULER number in the even suffix notation. If we assume somewhat less, namely that p -1 { n and n > r , then for p > 2, 1 = 1 we may take a, = B,/n in (l.l), where B, is the BERNOULLI number in the even suffix notation (see [ 5 ; Chap. 141).

From time to time congruences quite similar to (1.1) appear. Recently in 141 it was shown that the HERMITE polynomials H,(x) satisfy (modm") 'The coefficient of u,+,, vn+s is clearly