Answer to Question #350952 in Combinatorics | Number Theory for Fred

Question #350952

42. Prove that there exists an increasing infinite sequence of triangular 

numbers (i.e. numbers of the form tn = -1/2 n(n+ 1), n = 1, 2, ... ) such that 

every two of them are relatively prime. 


1
Expert's answer
2022-06-16T09:02:27-0400

We show first that if for some positive integer m we have m triangular 

numbers "a_1 < a_2 < \\dots < a_m" which are pairwise relatively prime, then there 

exists a triangular number "t>a_m" such that "(t, a_1, a_2,\\dots , a_m) = 1".


In fact, let "a = a_1 a_2 \\cdots a_m"; the numbers "a+ 1" and "2a+ 1" are relatively prime to "a". The number 

"a_{m+1}=t_{2a+1}=\\frac{(2a+1)(2a+2)}{2}=(a+1)(2a+1)"

is triangular number ">a_m" being relatively prime to a, it is relatively prime 

to every number "a_1,a_2,\\dots,a_m".


It follows that if we have a finite increasing sequence of pairwise relatively prime triangular numbers, then we can always find a triangular number exceeding all of them and pairwise relatively prime to them. Taking always the least such number we form the infinite sequence 

"t_1=1, t_2=3, t_4=10, t_{13}=91, t_{22}=253,\\dots"

of pairwise relatively prime triangular numbers. 


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