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 on the theory of algebraic number realms, published in volume 4 of the Jahreshericht der Deutschen Mathematiker-Vereinigung, and volume 1 of the Encyklopädie der mathematischen Wissenschaften.

The prize is offered under the following more particular conditions:

Among the additional stipulations it may be of interest to note that the prize will not be awarded before at least two years have elapsed since the first publication of the paper which is adjudged as worthy of the prize. In the meantime the mathematicians of various countries are invited to express their opinion as regards the correctness of this paper. The secretary of the Gesellschaft will write to the person to whom the prize is awarded and will also publish in various places the fact that the award has been made. If the prize has not been awarded before September 13, 2007, no further applications will be considered.

While this prize is open to the people of all countries it has become especially well known in Germany, and hundreds of Germans from a very noted university professor of mathematics to engineers, pastors, teachers, students, bankers, officers, etc., have published supposed proofs. These publications are frequently very brief, covering only a few pages, and usually they disclose the fact that the author had no idea in regard to the real nature of the problem or the meaning of a mathematical proof. In a few cases the authors were fully aware of the requirements but were misled by errors in their work. Although the prize was formally announced more than seven years ago no paper has as yet been adjudged as fulfilling the conditions.

It may be of interest to note in this connection that a mathematical proof implies a marshalling of mathematical results, or accepted assumptions, in such a manner that the thing to be proved is a necessary consequence. The non-mathematician is often inclined to think that if he makes statements which can not be successfully refuted he has carried his point. In mathematics such statements have no real significance in an attempted proof. Unknowns must be labeled as such and must retain these labels until they become knowns in view of the conditions which they can be proved to satisfy. The pure mathematician accepts only necessary conclusions with the exception that basal postulates have to be assumed by common agreement.

The mathematical subject in which the student usually has to contend most frequently with unknowns at the beginning of his studies is