Isn't there any divisor $k$ of $n^4$ such that $n^2-n<k<n^2$?

I did some experiment with my Python script to find a number which could divide $$n^4$$ in this interval ($$n^2-n$$, $$n^2$$). I watched the form of prim factors of the numbers in this ($$n^2-n$$, $$n^2$$) interval, but I hadn't was any idea for proofing it. Can anyone to help with some start idea to proof that if $$n^2-n < k < n^2$$ then $$k$$ can't divide $$n^4$$.

• $n$ doesn't divide $k$. Since $n<k$, there must be a prime number $p$ which divides $k$ but not $n$, thus also not $n^4$, so $k$ cannot divide $n^4$ – Pink Panther Mar 28 at 12:28
• @PinkPanther Why does the fact that $n$ does not divide $k$ imply that there is a prime $p$ dividing $k$ but not $n$? After all, $6$ does not divide $3$. – lulu Mar 28 at 12:31
• @lulu you have to use the fact that $n<k$, then it should be very straight forward. – Pink Panther Mar 28 at 12:32
• @PinkPanther Could you write it out? $12$ does not divide $18$ either... – lulu Mar 28 at 12:33
• @lulu you are right. I guess what I wanted to say is that there is a prime power which divides $k$ but not $n$, that is, there exists $p$ and $l$ such that $p^l$ divides $k$ but not $n$.I will write a complete answer in a moment. – Pink Panther Mar 28 at 12:37

Let $$0 and suppose that $$n^2-i$$ divides $$n^4$$. Then, $$n^4=(n^2-i)(n^2+j)=n^4+(j-i)n^2-i\,j\implies i\,j=(j-i)n^2$$ for some integer $$j$$. If $$j\le i$$, then $$n^4=(n^2-i)(n^2+j)\le(n^2-i)(n^2+i)=n^4-i^2.$$ Thus, $$j>i$$. Since $$i, it follows that $$j>(j-i)n\ge n$$. Moreover $$n\le j=\frac{i\,n^2}{n^2-i}<\frac{i\,n^2}{n^2-n}=\frac{i\,n}{n-1}\implies i>n-1,$$ a contradiction with the fact that $$i.

• Thank you for your nice answer, but I can't see one step at this moment: How can we now that $j>i$ ?. And this presumption is used in $(j−i)n>n$ as I see. – László Szilágyi Mar 28 at 16:21
• $ij > 0$ and $n^2 > 0$ so if $ij = (j-i)n^2 > 0$ then $j-i > 0$. ...(I suppose it should be points out that $i$ is positive (as $i = n^2 k; n^2 -n < k < n^2$. Ad that that $j > 0$ As if $k < n^2$ then $\frac {n^4}{k} > n^2$ so $n^2 + j = \frac {n^4}{k}$.) – fleablood Mar 28 at 17:14
• My answer is along the same lines, and I had deleted it when I saw that, but then I noticed that you claimed that $(j-i)n\gt n$ when it is not clear why $j-i\gt1$. – robjohn Mar 28 at 17:28
• @fleablood Why $i*j>0$ is ? – László Szilágyi Mar 28 at 18:25
• Why is $i*j > 0$. Seriously? Because $i > 0$ and $j > 0$. And, as I said $i > 0$ because $n^2 > k$ so $i = n^2 -k > 0$ and $j > 0$ because $k < n^2$ so $n^2 = \frac {n^4}{n^2} < \frac {n^4}k$. So $j = \frac {n^4}k - n^2 > 0$. – fleablood Mar 28 at 19:11

Let $$k=n^2-a$$. Since $$n^2-a\mid n^4$$ and $$\frac{n^4}{n^2-a}=n^2+a+\frac{a^2}{n^2-a}$$ we must have $$d=\frac{a^2}{n^2-a}\in\mathbb{Z}$$. However, since $$\color{#C00}{1\le a\le n-1}$$ and $$\frac{a^2}{n^2-a}$$ is increasing in $$a$$, $$0\lt\overbrace{\ \frac1{n^2-1}\ }^{\large\color{#C00}{a=1}}\le d\le\overbrace{\frac{n^2-2n+1}{n^2-n+1}}^{\large\color{#C00}{a=n-1}}\lt1$$ which is impossible because there is no integer between $$0$$ and $$1$$.

Bounding the Greatest Factor of $$\boldsymbol{n^4}$$ less than $$\boldsymbol{n^2}$$

Case: $$\boldsymbol{d=1}$$

Since $$n^2=a^2+a$$, we have $$n-1\lt a\lt n$$, which is impossible.

Case: $$\boldsymbol{d=2}$$

Since $$n^2=\frac{a(a+2)}2$$, $$\sqrt2\,n-1\lt a\lt\sqrt2\,n$$. In fact, $$2n^2+1=(a+1)^2$$. That means that $$\left(\frac{a+1}n\right)^2=2+\frac1{n^2}\lt\left(\sqrt2+\frac1{2n^2}\right)^2$$ Which forces $$\frac{a+1}n$$ to be a continued fraction overestimate of $$\sqrt2$$ . This gives the first pairs $$(n,a)$$ to be $$\{(0,0),(2,2),(12,16),(70,98),(408,576),(2378,3362),(13860,19600)\}$$ The recursion for $$n_k$$ is $$n_k=6n_{k-1}-n_{k-2}$$ and $$a_k=\left\lfloor\sqrt2\,n\right\rfloor$$.

In any case, we have

The largest factor of $$n^4$$ less than $$n^2$$ must be less than $$n^2-\sqrt2\,n+1$$ and there are an infinite number of $$n$$ so that the largest factor of $$n^4$$ less than $$n^2$$ is greater than $$n^2-\sqrt2\,n$$.

• Why $m-k>=1$ is ? – László Szilágyi Mar 28 at 18:26
• @LászlóSzilágyi: I have extended the answer to cover that and to free up $k$. – robjohn Mar 28 at 19:19
• I would also like to understand your derivation, but now I can see a few letters of error in it and then pleas correct it if you have the time. /: ($b-a=n^2/ab$) and what is role of $k$ – László Szilágyi Mar 28 at 21:49
• I changed from $k$ and $m$ to $a$ and $b$ to avoid conflict with $k$ from the question. In the first sentence, it is stated that $k=n^2-a$. Since $n^4=n^4+(b-a)n^2-ab$, we have $(b-a)n^2=ab$. I fixed the typo. – robjohn Mar 28 at 22:51
• @LászlóSzilágyi evidently the sharp result is that we can have a factor of $n^4$ as large as $n^2 - \lfloor n \sqrt 2 \rfloor \; , \;$ but no larger. This occurs with integers $x,y > 0,$ $x^2 - 2 y^2 = 1,$ and then $n=2xy.$ – Will Jagy Mar 29 at 0:19

The best examples come from $$n=2xy,$$ where $$x,y>0$$ are integers and $$x^2 - 2 y^2 = \pm 1 \; . \;$$ In these cases we get a factor $$m$$ of $$n^4$$ near $$n^2 - n \sqrt 2 \; . \;$$ When $$x^2 - 2 y^2 = 1,$$ we get $$m = n^2 - 2 x^2.$$ When $$x^2 - 2 y^2 = -1,$$ we get $$m = n^2 - 4 y^2.$$

jagy@phobeusjunior:~\$ ./mse
n: 2 =  2  m:  2  n^2 - m:  2 =  2
n: 3 =  3  m:  3  n^2 - m:  6 =  2 3
n: 4 =  2^2  m:  8  n^2 - m:  8 =  2^3
n: 6 =  2 3  m:  27  n^2 - m:  9 =  3^2
n: 10 =  2 5  m:  80  n^2 - m:  20 =  2^2 5
n: 12 =  2^2 3  m:  128  n^2 - m:  16 =  2^4
n: 24 =  2^3 3  m:  512  n^2 - m:  64 =  2^6
n: 30 =  2 3 5  m:  810  n^2 - m:  90 =  2 3^2 5
n: 60 =  2^2 3 5  m:  3456  n^2 - m:  144 =  2^4 3^2
n: 70 =  2 5 7  m:  4802  n^2 - m:  98 =  2 7^2
n: 84 =  2^2 3 7  m:  6912  n^2 - m:  144 =  2^4 3^2
n: 140 =  2^2 5 7  m:  19208  n^2 - m:  392 =  2^3 7^2
n: 168 =  2^3 3 7  m:  27783  n^2 - m:  441 =  3^2 7^2
n: 180 =  2^2 3^2 5  m:  32000  n^2 - m:  400 =  2^4 5^2
n: 408 =  2^3 3 17  m:  165888  n^2 - m:  576 =  2^6 3^2
n: 594 =  2 3^3 11  m:  351384  n^2 - m:  1452 =  2^2 3 11^2
n: 816 =  2^4 3 17  m:  663552  n^2 - m:  2304 =  2^8 3^2
n: 1170 =  2 3^2 5 13  m:  1366875  n^2 - m:  2025 =  3^4 5^2
n: 2378 =  2 29 41  m:  5651522  n^2 - m:  3362 =  2 41^2
n: 3230 =  2 5 17 19  m:  10425680  n^2 - m:  7220 =  2^2 5 19^2
n: 4756 =  2^2 29 41  m:  22606088  n^2 - m:  13448 =  2^3 41^2
n: 5880 =  2^3 3 5 7^2  m:  34560000  n^2 - m:  14400 =  2^6 3^2 5^2
n: 13860 =  2^2 3^2 5 7 11  m:  192080000  n^2 - m:  19600 =  2^4 5^2 7^2
n: 16296 =  2^3 3 7 97  m:  265531392  n^2 - m:  28224 =  2^6 3^2 7^2
n: 27720 =  2^3 3^2 5 7 11  m:  768320000  n^2 - m:  78400 =  2^6 5^2 7^2
n: 42672 =  2^4 3 7 127  m:  1820786688  n^2 - m:  112896 =  2^8 3^2 7^2
n: 57960 =  2^3 3^2 5 7 23  m:  3359232000  n^2 - m:  129600 =  2^6 3^4 5^2
n: 58206 =  2 3 89 109  m:  3387795864  n^2 - m:  142572 =  2^2 3 109^2
n: 80782 =  2 13^2 239  m:  6525617282  n^2 - m:  114242 =  2 239^2

• (+1) for the inspiration to prove the maximum. – robjohn Mar 29 at 6:20