Kummer’s Original Type Congruence Relation for the Universal Bernoulli Numbers

The aim of this paper is to give a congruence on universal Bernoulli numbers which congruence is the same type of Kummer’s original paper [K]. The remakable thing is the index of prime power that is the modulus of the congruence is half of the original one. We mention in this paper that this estimate is best possible. It is suprising fact for the author that the critical index is not less than half of the original. The motivation of this work is the investigation on generalized Bernoulli-Hurwitz numbers by the author himself [Ô]. Kummer’s original type congruences in [Ô] hold modulo the same power as the original one. When the author was working to get a proof of such the congruences in [Ô], he knew several resarches on universal Bernoulli numbers, especially Adelberg’s remarkable papers [A1] and [A2]. Reglettablly, Lemma 3.2.8 is not yet proved. This lemma is a variant of 3.2.1 which is already proved. The Lemma 3.2.8 might be true by several reason. The author hope the Lemma would be proved in the near future. The number theorists in Japan did not know the univesal Bernoulli numbers. The author expects that this research useful for the reserchers who are interested in Bernoulli numbers and Hurwitz numbers.