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A STUDY OF THE PROPERTIES OF INTEGRAL 



NUMBERS.* 



Francis E. Nipher. 



In the 1917 edition of A History of Elementary Mathe- 

 matics, pp. 32-3, Cajori refers to the work of Nicomachus, 

 who lived about A. D. 100, as follows : 



"He gives the following important proposition: All 

 cubical numbers are equal to the sum of consecutive odd 

 numbers. Thus, 8 = 2 3 = 3 + 5; 27 = 3 3 = 7 + 9 + 11; 

 64 = 4 8 = 13 + 15 + 17 + 19. ' ' 



It seemed probable that such interesting relations did 

 not end so abruptly as this statement might indicate, 

 although information along this line seems to be lacking 

 in modern text-books and mathematical dictionaries. 



A series of consecutive numbers and likewise of con 

 secutive even numbers yielded no results of this character 



It was found that a series of consecutive odd numbers 

 from unity to any odd number y satifies the equation 



1 + 3 + 5 + 7+ y = (k±j\* = N * 



(±±*) 



where N is the number of terms in the series. The table 

 of numbers here given shows the relation of these num- 

 bers. In Chrystals' Algebra, Vol. I, p. 

 467, it is stated that this proposition was TABLE *• 

 known to the Greek geometers. It is y \ 



stated as follows : ' ' The sum of any num- 

 ber of consecutive odd integers beginning 

 with unity is the square of their number. ' ' 

 A table of squares furnished the neces- 

 sary data for a study of the subject. It 

 was at once apparent that any number x 

 raised to any degree n where n is an even 

 number could be represented as the sum of an arithmeti- 



* Presented before the Academy of Science of St. Louis, Jan. 19, 1920. 



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