Natural numbers that can be expressed as the product of two and three consecutive natural numbers

Question

Looking for natural numbers larger than 210 which can be expressed as the product of two continuous natural numbers or the product of three continuous natural numbers at the same time.

$6 = 1 \cdot 2 \cdot 3 = 2 \cdot 3$
$210 = 5 \cdot 6 \cdot 7 = 14 \cdot 15$

I have made a program to find that numbers smaller than $1000000000$, only to find $6$ and $210$. Are there any other ones?

Do you mean consecutive natural numbers? Because continuous describes certain functions, usually involving real or complex numbers. — Sammy Black, Mar 13 '21 at 07:29
The word you are looking for to describe a number, the next number, etc, is called consecutive — ndhanson3, Mar 13 '21 at 07:30
I do not understand your question, have you tried multiplying some continuous natural numbers — Aderinsola Joshua, Mar 13 '21 at 07:32
I think you're trying to find pairs of natural numbers $(m, n)$ such that $m(m+1)(m+2) = n(n+1)$. — Sammy Black, Mar 13 '21 at 07:32
@Sammy Black I am sorry because of my poor English,in china I can't find a website only communicating math problems,luckily my question was solved.Thank you all. — shenyuantao, Mar 13 '21 at 13:13

Parcly Taxel · Accepted Answer · 2021-03-13T07:44:26.167

This is equivalent to finding solutions of the elliptic curve Diophantine equation $$y(y+1)=y^2+y=x^3-x=(x+1)x(x-1)$$ which is 37.a1 in LMFDB. Obviously $x,y>0$; there are nevertheless only a finite number of integral solutions, which are listed in LMFDB. With the restriction on $x,y$ there are only two solutions: $$x=2,y=2\implies N=6$$ $$x=6,y=14\implies N=210$$ Hence $6$ and $210$ are the only numbers of the form you mentioned.

Natural numbers that can be expressed as the product of two and three consecutive natural numbers

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