{"title":"Conjectures as to a factor of 2 P ± 1","authors":"N. Y. Wilson","doi":"10.1017/S0950184300003086","DOIUrl":"https://doi.org/10.1017/S0950184300003086","url":null,"abstract":"More than 300 years ago Mersenne made pronouncements as to the prime or composite nature of 2 — 1 for all prime values of p from 1 to 257. His reasons were not disclosed, but the combined efforts of many mathematicians have shown that his statements were substantially correct. Of course when p is composite, two or more factors of 2 ± 1 will often be obvious but, as far as I am aware, only two general theorems relating to a non-obvious factor have been proved. I give these by way of introduction to this article and thereafter proceed to enunciate seven new and original theorems. As I offer no theoretical proofs of the theorems, they must be regarded as conjectures. They are, however, based on extensive computation which has engaged me for a period of three years.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"316 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116102873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"“The equal internal bisectors theorem, 1840-1940. … Many solutions or none?” A centenary account","authors":"J. A. M'Bride","doi":"10.1017/S0950184300000021","DOIUrl":"https://doi.org/10.1017/S0950184300000021","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132224246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On an approximate construction for a regular polygon","authors":"S. Scott","doi":"10.1017/S0950184300002901","DOIUrl":"https://doi.org/10.1017/S0950184300002901","url":null,"abstract":"My attention was drawn by an Art teacher to the following approximate construction for inscribing a regular polygon of n sides in a given circle, having diameter AB , centre O . Find C in AB so that AC : AB = 2: n , and construct the equilateral triangle ABD . If DC produced meets the circle in E , then AE is approximately a side of the required polygon.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133861894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Riemann surfaces of a function and its fractional integral","authors":"W. Fabian","doi":"10.1017/S0950184300003116","DOIUrl":"https://doi.org/10.1017/S0950184300003116","url":null,"abstract":"2. Transformation of Riemann surfaces. Theorem 1. Let f(z) be analytic within a circle with centre at a, and which contains I in its interior. Then a is a branch-point of D~ (la)f(z) for non-integral values of X. If A is a rational fraction rjg expressed in its lowest terms, then a is the vertex of a cycle of s roots. If A is irrational or complex, then a is the vertex of an infinite number of roots.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115778750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A dual quadratic transformation associated with the Hessian conics of a pencil","authors":"T. Scott","doi":"10.1017/S0950184300002652","DOIUrl":"https://doi.org/10.1017/S0950184300002652","url":null,"abstract":"1. The invariants and covariants of a system of two conics have been much studied 2 but little has been said about those of three conies. Three conics have a symmetrical invariant Ω 123 , or in symbolical notation ( a b c ) 2 . According to Ciamberlini 3 the vanishing of this invariant signifies that the Φ conic of any two of f 1 , f 2 , f 3 is inpolar with respect to the third; and in a previous paper 4 I have derived by symbolical methods a more symmetrical result, viz., if Ω 123 vanishes, then u being any line in the plane, u 1 , u 2 , u 3 are concurrent, where u i is the polar with respect to f i of the pole of u with respect to Φ jk .","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115790311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The characteristic polynomials of AB and BA","authors":"J. Williamson","doi":"10.1017/S0950184300003104","DOIUrl":"https://doi.org/10.1017/S0950184300003104","url":null,"abstract":"Let A and B be square matrices of the same order, with elements in any field F; it is well known that the characteristic polynomials of AB and BA are the same (see, e.g. C. C. Macduffee, Theory of Matrices, p. 23). The proof of this is easy when one at least of the matrices is non-singular; the object of the following remarks (which are not claimed as original) is to point out that the case | A | = | B | = 0 is just as easy. If one attempts to deduce the result in this case from the result in the non-singular case, unnecessary restrictions on the field F are apt to appear (see e.g., W. V. Parker, American Mathematical Monthly, vol. 60 (1953) p. 316). If one proceeds directly to the general case, no difficulties are encountered. Consider the elements of A as indeterminates over F; then ABA-XI = | ABA -XA = AB XI A , the equality holding in the sense of an identity between polynomials in ou, o12 , ann, A, with coefficients in F. Since j A | , as a polynomial in the elements of A, is not zero, we may divide by it to obtain | BA — XI | = | AB — XI | . This relation must still be true (as between polynomials in A with coefficients in F) when we regard the elements of A as being fixed members of F; this is the required result. We have used here only the results that the determinant of the product of two square matrices is equal to the product of their determinants (the usual proofs cover the present case without alteration), and that the product of two polynomials, in any indeterminates with coefficients in a field, cannot be zero unless one of the factors is zero. Both these results are quite elementary. A variation of the above reasoning, which minimises the number of indeterminates, is to consider the matrix (A — /J.1) B (A — fil) —A (A — fil), where now the elements of A are fixed members of Ft and A and ft are indeterminates. The argument is as before, and we put finally /x = 0.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121028927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Definition of a Tangent to a Curve","authors":"T. M. Flett","doi":"10.1017/S0950184300003153","DOIUrl":"https://doi.org/10.1017/S0950184300003153","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131002881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"John G. B. Meiklejohn: 1872–1951","authors":"H. Jack","doi":"10.1017/S0950184300003025","DOIUrl":"https://doi.org/10.1017/S0950184300003025","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133382056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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