SQUARE NUMBERS. 



(10) Modulus 17. 

 (17/ ± ^) '' i I77«' = '^' 



(•7;> ± ?) '^ ± 17 y«' = ■"'^ 



(i7/> + 6) /' ± 17?"' = "^ 



{17/. + 7) /'- ± 17?"' = •«'" 



A great variety of impoffible forms might have been 

 given to other moduli ; but the above are fufficient for our 

 prefent purpofc. 



There arc alfo many formula:, which, though polhble 

 fingly, become impoffible in pairs : fuch are the foUowmg : 



H 



Thefe might alfo be carried to a much greater extent, 

 and many collateral properties drawn from them relative to 

 the impoffibihty of fonie higher powers : we mult not, how- 

 ever, carry the fubjeft farther in this place. The reader, 

 who is defirous of more detailed information, may confult 

 Barlow's " Elementary InvefUgation of the Theory tjf Num- 

 bers," where this part of the doftrine of numbers is carried 

 to a confiderable extent. We (hall merely feleft a few 

 other dillindl properties of fquares, as they are given by 

 the fame author, in his " Mathematical DiAionary." 



17. The fum of two odd fquares cannot be a fquare. 



18. An odd fquare, taken from an even fquare, cannot 

 leave a fquare remaicder. 



19. If the fum of two fquares be itfelf a fquare, one of 

 the three fquares is divifible by 5. 



20. Square numbers muft terminate in one of the digits 

 O, I, 4, 5, 6, or 9. 



21. No number of repetend digits can be a iquarc. 



22. The area of a rational right-angled triangle cannot 

 be equal to a fquare. 



23. The two following feries are remarkable for being 

 fuch, as, when reduced to improper fraftions, the fum of 

 the fquares of each numerator and denominator is a com- 

 plete fquare ; or, which is the fame, they are the fides of 

 rational right-angled triangles. Thefe feries are as fol- 

 low, viz. 



U. 2|i 3f. 4t. SrV> &c. &c. 

 »ii 2^T. %]h 44« 514. &c- &c. 



24. The fecond differences of confecutive fquare numberivj 

 are equal to each other, thus : 



Squares 1 



Firll difference 

 Second difference 



, 4, 9, 16, 25, &c. 



3> 5' 7. 9» &c. 

 2, 2, 2, &c. 



To thefe we may alfo add the following ; which are more 

 particularly applicable to the indetermmate and Diophantine 

 analyfis. 



25. If a number be the fum of two fquares, its double is 

 alfo the fum of two fquares ; for 



(.,'-+ j') X 2= {x +y)'k- i^'-yY- 



Hence alfo, the fum of two fquares multiplied by any 

 power of 2, is the fum of two fquares. 



26. The produft of two numbers, each being the fum of 

 two fquares, is itfelf the fum of two fquares ; for 



(■' 



') X (.v' 



+ y'') 

 .1. 



iix.'+yy'y+{xy'-x>yy, 



(.v-r' 



Thus, 



5 = 2 

 13 = 3 



+ 



Produd 65 = 8' -t- !■ 



7' + 4' 



27. The produft of the fum of four fquares, by the fum 

 of four other fquares, is itfelf the fum of four fquares ; thuj 



(iv^ + .v^ + y' + z') X {w'^ + x" + y +»'•) = 

 '[•wiu' -f xx' + yy' -f- zz')^ -{- (iw.v' — xtv' 



+ y^' - y'z)' + 

 (y' lu — .vz' — y'w'+ z-v')" -f (wz' + xy' 

 — y x' — X ■»') ' 



as will appear by the developement of thefe formulae. 



28. Every integral number is either a fquare, or the funj 

 of two, three, or four fquares. 



The latter is one of the celebrated numerical theorems of 

 Fcrmat, which was firft demonftrated by Lagrange. 



For a variety of other properties, fee the works above 

 referred to. 



We fliall conclude this article with a table of the fquares 

 and cubes of all numbers from i to 1200. A table of the 

 fquare and cube roots, to the fame extent, is given under 

 the article Root. 



Table 



