Oblong numbers : x(x+ 1), y(y+ 1), z(z+ 1) in arithmetic progression

 

An Oblong number is a number which is the product of two consecutive integers, that is, a number of the form   n \,(n + 1)

 
 
There exist infinitely many triplets of positive integers   x, \; y, \; z,   for which the numbers
       x \,(x+1),     y \,(y+1),     z \,(z+1)
form an increasing arithmetic progression.

The required property holds for

(1)

x \; = \; n
y \; = \; 5 \, n \; + \; 2
z \; = \; 7 \, n \; + \; 3

n   is a positive integer

since in this case the numbers

n \,(n+1),     (5 \, n+2) \,(5 \, n+3),     (7 \, n+3) \,(7 \, n+4)

form the arithmetic progression with the common difference   6 \, (2 \, n+1)^2

(5 \, n+2) \,(5 \, n+3) \; - \; n \,(n+1)
= \; (25 \, n^2 + 25 \, n + 6) \; - \; (n^2 + n)
= \; 24 \, n^2 \; + \; 24 \, n \; + \; 6
= \; 6 \, (2 \, n+1)^2

(7 \, n+3) \,(7 \, n+4) \; - \; (5 \, n+2) \,(5 \, n+3)
= \; (49 \, n^2 + 49 \, n + 12) \; - \; (25 \, n^2 + 25 \, n + 6)
= \; 24 \, n^2 \; + \; 24 \, n \; + \; 6
= \; 6 \, (2 \, n + 1)^2

 

and for

(2)

the numbers  

x \; = \; n,  
y \; = \; 29 \,n \; + \; 14,
z \; = \; 41 \,n \; + \; 20  

form an arithmetic progression:

(29 \, n+14) \,(29 \, n+15) \; - \; n \,(n+1) \; = \; 210 \, (2 \, n + 1)^2
(41 \, n+20) \,(41*n+21) \; - \; (29 \, n+14) \,(29 \, n+15) \; = \; 210 \, (2 \, n + 1)^2

(29 \, n+14)(29 \, n+15) - n \,(n+1) = (41 \, n+20)(41 \, n+21) - (29 \, n+14)(29 \, n+15)

 

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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