Answer to Question #350953 in Combinatorics | Number Theory for Fred

Question #350953

43. Prove that there exists an increasing infinite sequence of tetrahedral 

numbers (i.e. numbers of the form Tn = 1/6 n(n+ 1)(n+2), n = 1,2, ... ), such 

that every two of them are relatively prime. 


1
Expert's answer
2022-06-16T09:05:09-0400

We shall prove first that if for some positive integer m the tetra-hedral numbers "a_1<a_2<\\dots<a_m" are pairwise relatively prime, then there exists a tetrahedral number "T > a_m" such that "(T, a_1, a_2, ... , a_m) = 1" . In fact, let "a = a_1a_2 \\dots a_n". Put "T = T_{6a+1} = (6a+1) (3a+1) (2a+1)"; clearly "T" is prime relatively to "a", hence relatively to each of the numbers "a_1, \\dots , a_m", and "T>a\\geq a_m".


Thus, we can define the required increasing sequence of pairwise relatively prime tetrahedral numbers by induction: take "T_1=1" as the first term of the sequence, and, after having defined "m" first pairwise relatively prime tetra-hedral numbers of this sequence, define the "m+1st" as the least tetra-hedral number exceeding all first m terms, and being relatively prime to each of them. In this manner we obtain the infinite increasing sequence of pairwise relatively prime tetrahedral numbers 

"T_1 =1, T_2 = 4, T_5 = 35, T_{17} = 969,\\dots."


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS