From: Bill Dubuque Newsgroups: sci.math Subject: sloppy proofs everywhere: extremal errors; 1 is prime? [was: Pertti Lounesto's challenge: Invalidate my counterexamples] Date: 29 Dec 1997 03:33:01 -0500 Pertti Lounesto wrote to sci.math: | | Invalidate the counterexamples presented at my www-page with URL: | "http://www.hit.fi/~lounesto/counterexamples.htm". | | In this www-page I give 30-40 counterexamples to proven theorems, | published in recent mathematical papers ... Ron Bloom replied: : I would be interested to know if you have been able, working backwards : from a counterexample, identify the logical error in the proof(s) : of the theorem which a particular counterexample invalidates. Pertti replied: | In some cases yes, but not always. Most often the case is that the | author has not considered some special case (e.g. a product is 0), | or is just guessing on the basis of what happens in low dimensions. There are plenty of analogous "extremal errors" in number theory and algebra textbooks, especially having to do with whether 1 is considered prime or not (and related instances such as whether the entire ring is allowed as a prime ideal, or the one element ring is allowed as an integral domain, etc). Even the well-known classic time-tested textbook on number theory by Hardy and Wright commits such errors: e.g. though they employ the modern definition that 1 is not prime (p. 2), they later state the Lucas primality test (the simple converse of Fermat's little Theorem FlT) in a form that implies that one is prime (p. 72): THEOREM 90. If a^(m-1) = 1 (mod m) and a^x != 1 (mod m) for any divisor x of m-1 less than m-1, then m is prime. If you flip through the pages of almost any number theory textbook and check the case p=1 (or analogous extremal cases) you will almost surely find errors like "Hardy's Theorem 90" (apologies to Hilbert!). Analogous problems also occur for the "oddest" prime p=2, a situation sometimes referred to as the "terrible 2s", or "the trouble with 2". Perhaps in the old days p=1 was more properly the "oddest" prime! This sloppy practice of stating "generic" results will probably disappear in the future when automatic proof checkers are routinely used to check math texts (especially considering that the extremal cases usually have much simpler proofs than the generic case). Landau's "Grundlagen" has already been checked in the Automath system [1],[2],[3],[4]. It would be interesting to do the same for Hardy and Wright. Note also that threads on the topic of the primality of 1 often appear both here on sci.math and also in the MAA math-history-list, archived at http://forum.swarthmore.edu/epigone/math-history-list/ It would be interesting to consider the historical evolution of this convention in all its equivalent forms, including the ideal theoretic forms: 1 is disallowed as a prime for many more reasons than unique factorization and it would be interesting to study the historical interplay between these various motivations. Based upon replies in earlier threads it would appear that currently no such historical study exists. Perhaps fragments of such a study are currently intermingled in other works. Anyone? -Bill Dubuque [1] van Benthem Jutting, L. S. Checking Landau's "Grundlagen" in the Automath system [MR 58 #32124ab]. in [2]. CMP 1 429 425 03B35 [2] Selected papers on Automath. Edited by R. P. Nederpelt, J. H. Geuvers and R. C. de Vrijer with the assistance of L. S. van Benthem Jutting and D. T. van Daalen. Studies in Logic and the Foundations of Mathematics, 133. North-Holland Publishing Co., Amsterdam, 1994. xx+1024 pp. ISBN: 0-444-89822-0 CMP 1 429 395 03B35 (03-06 03B40) [3] de Bruijn, N. G. A survey of the project AUTOMATH. To H. B. Curry: essays on combinatory logic, lambda calculus and formalism, pp. 579--606, Academic Press, London-New York, 1980. MR 81m:03017 (Reviewer: Frank Malloy Brown) 03B35 (03B40 68G15) [4] van Benthem Jutting, L. S. Checking Landau's Grundlagen in the AUTOMATH system. Doctoral dissertation, Technische Hogeschool Eindhoven, Eindhoven, 1977. With a Dutch summary. Technische Hogeschool Eindhoven, Eindhoven, 1977. v+121 pp. MR 58 #32124a (Reviewer: I. Kramosil) 68A40 (02-04 68A30)