Citation Context

...ions of (5) are given by (8) ~yt + /x 2 - \zi = (yQ + /x 2 - lz0) (x + /x 2 - l) 1 i = 0, 1, 2, . . . , where (yo,Zo) can take only a finite number of values, one of which must be (2>,1). (See Nagell =-=[6]-=-.) There is a one-to-one correspondence between the solutions (y-c^z^) of (4) and (^{,2£) of (5) where ~yi = byi because ~y\ = b(b - a + az\) , and hence y ^ is always an integer. Tk&oKtm 7: Let , (9)...

Citation Context

...INTRODUCTION A P-set will be defined as a set of positive integers such that if a and b are two distinct elements of this set, ab + 1 is a square. There are many examples of P-sets such as [2, 12] or =-=[1, 3, 8, 120]-=- and even formulas such as or [n - 1, n + 1, 4n, 4n(4n 2 - 1)] [77?, n 1 - 1 + (m - 1) (n - l ) 2 , n(nm + 2 ) , kmirrm 1 - ran + In - l ) 2 + 4(rnn 2 - run + In - 1)]. (See Cross [1].) However, none ...

Citation Context

... Canada 1. INTRODUCTION A P-set will be defined as a set of positive integers such that if a and b are two distinct elements of this set, ab + 1 is a square. There are many examples of P-sets such as =-=[2, 12]-=- or [1, 3, 8, 120] and even formulas such as or [n - 1, n + 1, 4n, 4n(4n 2 - 1)] [77?, n 1 - 1 + (m - 1) (n - l ) 2 , n(nm + 2 ) , kmirrm 1 - ran + In - l ) 2 + 4(rnn 2 - run + In - 1)]. (See Cross [1...

Citation Context

...wever, none of these formulas are general. More recently, there has been considerable work on P-sets with polynomials (by Jones [2, 3]) and in connection with Fibonacci numbers (by Hoggatt and Bergum =-=[4]-=-). It is of interest to find out how much these sets can be extended by adding new positive integers to the set; for example [2, 12] can be extended to [2, 12, 420], A P-set which cannot be extended w...

Citation Context

... ate/1/ 3 + (ab + be + oa)N 2 + (a + b + o)N + 1 = t 2 . Let # = max{ate, ab + bo + oa, a + b + o]. Now, as abcN 3 + (ab + bo + oa)N 2 + (a + b + £)/!/ + 1 has no squared linear factor in N, by Baker =-=[7]-=-, N < exp{(10 6 #) 1()6 }. Until recently there was no way of knowing if a P-set was nonextendible if it had four elements. However, in Baker and Davenport [8], it has been proved that [1, 3, 8, 120] ...

Citation Context

... [8] would seem workable for checking if there are other sets of four which are nonextendible.272 THE STUDY OF POSITIVE INTEGERS (a,b) [Oct. A recent adaptation of this method was given by Grinstead =-=[9]-=-. 4. RESTRICTIONS ON EXTENDING FOUR-MEMBER P-SETS First it should be noted that Baker and Davenport [8] (from their relationship (20) and Section 5) seem to indicate that any fifth member of a Pset th...