Prove that the product of four consecutive positive integers plus one is a perfect square

Question

I need to prove the following, but I am not able to do it. This is not homework, nor something related to research, but rather something that came up in preparation for an exam.

If $n = 1 + m$, where $m$ is the product of four consecutive positive integers, prove that $n$ is a perfect square.

Now since $m = p(p+1)(p+2)(p+3)$;

$p = 0, n = 1$ - Perfect Square

$p = 1, n = 25$ - Perfect Square

$p = 2, n = 121$ - Perfect Square

Is there any way to prove the above without induction? My approach was to expand $m = p(p+1)(p+2)(p+3)$ into a 4th degree equation, and then try proving that $n = m + 1$ is a perfect square, but I wasn't able to do it. Any idea if it is possible?

Answer 1

Your technique should have worked, but if you don't know which expansions to do first you can get yourself in a tangle of algebra and make silly mistakes that bring the whole thing crashing down.

The way I reasoned was, well, I have four numbers multiplied together, and I want it to be two numbers of the same size multiplied together. So I'll try multiplying the big one with the small one, and the two middle ones.

$$p(p+1)(p+2)(p+3) + 1 = (p^2 + 3p)(p^2 + 3p + 2) + 1$$

Now those terms are nearly the same. How can we force them together? I'm going to use the basic but sometimes-overlooked fact that $xy = (x+1)y - y$, and likewise $x(y + 1) = xy + x$.

$$\begin{align*} (p^2 + 3p)(p^2 + 3p + 2) + 1 &= (p^2 + 3p + 1)(p^2 + 3p + 2) - (p^2 + 3p + 2) + 1 \\ &= (p^2 + 3p + 1)(p^2 + 3p + 1) + (p^2 + 3p + 1) - (p^2 + 3p + 2) + 1 \\ &= (p^2 + 3p + 1)^2 \end{align*}$$ Tada.

Answer 2

$(n-1)(n+1)+1 = n^{2}$.

Note that $(n+1)-(n-1)=2$.

With this in mind

$$\begin{align*} p(p+1)(p+2)(p+3)+1 &= (p^{2}+3p)(p^{2}+3p+2)+1 \\ &= [(p^{2}+3p+1)-1][(p^{2}+3p+1)+1]+1 \\ &= (p^{2}+3p+1)^2 \end{align*}$$

Answer 3

Here's another way which begins by exploiting a symmetry in the expression.

Notice that if you substitute $x=p+\frac{3}{2}$, the expression becomes

$$\left(x-\frac{3}{2}\right)\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)\left(x+\frac{3}{2}\right) + 1$$

Now see that the terms make the product of 2 differences of squares

$$\begin{align} & \quad \left(x+\frac{3}{2}\right)\left(x-\frac{3}{2}\right)\left(x+\frac{1}{2}\right)\left(x-\frac{1}{2}\right) + 1 \\&= \left(x^2-\frac{9}{4}\right)\left(x^2-\frac{1}{4}\right) + 1 \\ &= \left(x^4 - \frac{10}{4} x^2 + \frac{9}{16}\right) + 1 \\ &= x^4 - \frac{10}{4} x^2 + \frac{25}{16} \\ &= \left(x^2 - \frac{5}{4}\right)^2 \\ &= \left(p^2 + 3p + 1\right)^2 \end{align}$$

which is a perfect square.

Answer 4

Below I present a generalization. $ $ Using the abbreviations$\rm\ \ c = a\!+\!b,\ \ \color{red}d = ab/2\:\ $ we compute

$$\rm\begin{eqnarray} &&\rm\qquad\quad\ \color{blue}{(x\!+\!a)\,(x\!+\!b)}\,(x\!+\!c)\,x &=&\rm\, \color{blue}{(x^2\!+cx\ \ +\ \ ab\ \ \ \, )}\,(x^2+cx\:\!) \\ && &=&\rm\, (x^2\!+cx+d\ \,\color{red}{+\, d})\,(x^2+cx+d\, \color{red}{-\,d}) \\ && &=&\rm\, (x^2\!+cx+d)^2\! \color{red}{- d^2} \\ \rm b=2\quad &\Rightarrow&\rm\quad\ \ \ \ x(x+a)(x\!+\!2)(x\!+\!a\!+\!2) &=&\,\rm (x^2\!+(a\!+\!2)\,x+a)^2 -a^2 \\ \rm a=1\quad &\Rightarrow&\rm\qquad\quad\ \ \ x(x\!+\!1)(x\!+\!2)(x\!+\!3) &=&\rm\, (x^2\!+3\:\!x+1)^2 -1\ \ \ as\ sought. \end{eqnarray}$$

Answer 5

Set $p+1.5=q$. Now $$ \begin{align*}m &= (q-1.5)(q-0.5)(q+0.5)(q+1.5)+1 \\ &= (q-1.5)(q+1.5)(q-0.5)(q+0.5)+1 \\ &= (q^2 - 2.25)(q^2-0.25)+1 \\ \end{align*}$$

Let $q^2 = r$.

$$\begin{align*} m &= (r-2.25)(r-0.25)+1\\ &= r^2-2.5a+1.5625 \\ &= (r-1.25)^2. \end{align*}$$

This is a perfect square since r ends in 0.25 as q ends in 0.5

Basically the substitution converted it from a fourth degree to a quadratic which made it easy to deal with.

Answer 6

$$1+p(p+1)(p+2)(p+3)=1+ \dots +p^4.$$

If you want a general formula, it must be a square either of the form $(p^2+cp+1)^2$ or $(p^2+cp-1)^2$ for some constant $c$.

Expand the squares and the original product and match up two terms to calculate $c$. Verify that the other coefficients are correct as well.

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Details:

The expansion of the product is $p^4+6p^3+11p^2+6p+1$.

The expansions of the squares are $p^4 + 2cp^3+c^2p^2\pm2p^2\pm2cp+1$.

Comparing the coefficients of $p^3$ gives $c=3$ which evidently works with the plus sign, so we get $(p^2+3p+1)^2$.

Answer 7

If I am missing something I will take this answer down, but the following seems responsive to your question.

If

$m = 1 + x(x+1)(x+2)(x+3)$ we can expand this as $1+6x+11x^2+6x^3+x^4$.

This is

$m = (1+3x+x^2)^2$

So when x is an integer, this shows that m is a perfect square, without induction.

Answer 8

I think there are two issues here. One is constructing the quartic, which just depends on you doing the algebra correctly. The second is proceeding to factorise the quartic. It would be easier to factorise it if you know what the factorisation is going to be.

To discover this, I tried a few examples. For $p=7$, the quartic gives $5041=71^2$. For $p=14$, the quartic gives $57121=239^2$. I noticed that $71=72-1=8\times9-1$ and $239=15\times16-1$.

This suggested that the quartic was $((p+1)(p+2)-1)^2$.

Once you know the answer, it is easy to find it!

Answer 9

Take $p^2$ common after multiplying.
Then put $p +1/p =y$ and solve.

You will get the answer.

Answer 10

I have to add what I think is a 'dumb' way to do it by hand (with paper) as opposed to Alex B. succinct cleverness:

First, multiply out the product to get $p^4 + 6p^3 + 11p^2 + 6p + 1$.

Since this is a square, it must be a quadratic $p^2 + x p + y$.

Squaring the quadratic, ignoring a lot of the cruft, and just looking at just the second and last coefficients

$$6 = x + x$$

and

$$1 = y^2$$

and you're done.

Answer 11

Select any $a\in\mathbb{Z}_{\ge 2}$ and define $P$ to be the product of the four consecutive integers $a-1,a,a+1$ and $a+2$, that is $$P=(a-1)a(a+1)(a+2).$$ Expanding $P$, we get $$P=a^4+2a^3-a^2-2a.$$ Thus, we have $$P+1=a^4+2a^3-a^2-2a+1=(a^2+a-1)^2,$$ that is $P+1$ is a perfect square. Now since $a\in\mathbb{Z}_{\ge 2}$ is arbitrary, implies that $P+1$ is a perfect square for all $a\in\mathbb{Z}_{\ge 2}$. This completes the proof.

Answer 12

Expanding $p(p+1)(p+2)(p+3)+1$ we get

$$p^4+6p^3+11p^2+6p+1$$

Note that $p^4$ is $(p^2)^2$, so this is equal to a square plus something extra. If this is to be a square number, then the extra must be a sum of odd numbers starting with $2p^2+1$, that is, we must have

$$6p^3+11p^2+6p+1=\sum^n_{k=0}(2(p^2+k)+1)$$

for some $n$. That sum can be computed to be

$$(n+1)(2p^2+1)+(n+1)n$$

and now it's quite easy to see that there's only one possible choice for $n$. Indeed, we want this to be a cubic in $p$, so $n$ must be linear in $p$. The coefficient of the linear term must be $3$, so that we get a cubic term of $6p^3$ after multiplying out. And since we want a constant term of $1$, we see that there can be no constant term in $n$. So $n=3p$ is the only possible choice. Plugging this in, we find that it works.

Answer 13

Group the factors like this:

$p(p+3) \times (p+1)(p+2) + 1$

You should then expand to get $(x^2+3x)(x^2+3x+2) + 1$, which can be rewritten as $(x^2+3x+1-1)(x^2+3x+1+1) + 1$

This looks familiar, as the factors on the lefthand side is a difference of squares: $(x^2+3x+1)^2 - 1 + 1$

Cancel out the minus one and plus one and you have $(x^2+3x+1)^2$, which is a perfect square.

Answer 14

To solve this we can use the following theorem:-
Theorem:- For any arbitrary integer a, the following holds true:- $2|a(a+1)$.
Proof:- We know that a can either can be of the from $2q$ or $2q+1$. Therefore:-
i) If a = $2q$, then $a(a+1) = 2q(2q+1)$, which is divisible by 2.
ii) If a = $2q+1$, then $a(a+1) = (2q+1)(2q+2) = 2(q+1)(2q+1)$, which is divisible by 2.

Coming to our question we see that $a(a+1)(a+2)(a+3)+1$ is equivalent to $(2q)(2q')+1$ where $2q = a(a+1)$ and $2q'=(a+2)(a+3)$. Thus our expression turns out to be $4x+1$ where $x=qq'$. But we know that a odd perfect square is always of the form $4k+1$. Thus is it proved that $a(a+1)(a+2)(a+3)+1 = d^2$ for any integer d.

Answer 15

Recursion on the square root, k(n)

n = lead integer

k(n) = square root [n(n+1)(n+2)(n+3) + 1]

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k(1)^2 = 25

k(2)^2 = 121

k(3)^2 = 361

k(4)^2 = 841

k(5)^2 = 1681

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k(1) = 5

k(2) = 11 = 5+ 6 = 5 + 2*(2+1)

k(3) = 19 = 11+ 8 = 11 + 2*(3+1) = 5 + 2 * ((2+1) + (3+1))

k(4) = 29 = 19+10 = 19 + 2*(4+1) = 5 + 2 * ((2+1) + (3+1) + (4+1))

k(5) = 41 = 29+12 = 29 + 2*(5+1) = 5 + 2 * ((2+1) + (3+1) + (4+1) + (5+1))

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k(n) = 5 + 2 * ( (n+1)(n+2)/2 - (1+2) )

 = (n+1)(n+2) - 1

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induction step

k(n+1) = k(n) + 2(n+2)

   = (n+1)(n+2) - 1   + 2(n+2)

   = (n+2)(n+3) - 1

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k(n)^2 - 1 = ((n+1)(n+2))- 1)^2 - 1

        = ((n+1)(n+2))^2 - 2(n+1)(n+2)

        = ((n+1)(n+2)) * ((n+1)(n+2) - 2)

        = ((n+1)(n+2)) * (n^2 + 3n)

        = n(n+1)(n+2)(n+3)

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!!!

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Thinking a little more about k(n).

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A rough approximation of the root of ( n(n+1)(n+2)(n+3) + 1 ) would be n^2.

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We might improve it a bit by averaging the products of the outer and inner pair of factors,

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k(n) = (n(n+3) + (n+1)(n+2)) / 2

   = (n+1)(n+2) -1

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Dead on. Surprising, no?

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