Smallest Triplet of Consecutive Integers each Divisible by Fourth Power
Theorem
This triplet of consecutive integers has the property that each of them is divisible by a fourth power:
- $33 \, 614, 33 \, 615, 33 \, 616$
This is the smallest such triplet.
Proof
\(\ds 33 \, 614\) | \(=\) | \(\ds 14 \times 7^4\) | ||||||||||||
\(\ds 33 \, 615\) | \(=\) | \(\ds 415 \times 3^4\) | ||||||||||||
\(\ds 33 \, 616\) | \(=\) | \(\ds 2101 \times 2^4\) |
Each number in such triplets of consecutive integers is divisible by a fourth power of some prime number.
Only $2, 3, 5, 7, 11, 13$ are less than $\sqrt [4] {33 \, 616}$.
Case $1$: a number is divisible by $13^4$
The only multiple of $13^4$ less than $33 \, 616$ is $28 \, 561$, and:
\(\ds 28 \, 559\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 28 \, 560\) | \(=\) | \(\ds 2^4 \times 3 \times 5 \times 7 \times 17\) | ||||||||||||
\(\ds 28 \, 562\) | \(=\) | \(\ds 2 \times 14281\) |
Since neither $28 \, 559$ nor $28 \, 562$ are divisible by a fourth power of some prime number, $28 \, 561$ is not in a triplet.
$\Box$
Case $2$: a number is divisible by $11^4$
The only multiples of $11^4$ less than $33 \, 616$ are $14 \, 641$ and $29 \, 282$, and:
\(\ds 14 \, 639\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 14 \, 640\) | \(=\) | \(\ds 2^4 \times 3 \times 5 \times 61\) | ||||||||||||
\(\ds 14 \, 642\) | \(=\) | \(\ds 2 \times 7321\) | ||||||||||||
\(\ds 29 \, 281\) | \(=\) | \(\ds 7 \times 47 \times 89\) | ||||||||||||
\(\ds 29 \, 283\) | \(=\) | \(\ds 3 \times 43 \times 227\) |
Hence none of these numbers is in a triplet.
$\Box$
Case $3$: a number is divisible by $7^4$
There are $14$ multiples of $7^4$ less than $33 \, 616$, and:
\(\ds 2399\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 2400\) | \(=\) | \(\ds 2^5 \times 3 \times 5^2\) | ||||||||||||
\(\ds 2401\) | \(=\) | \(\ds 7^4\) | ||||||||||||
\(\ds 2402\) | \(=\) | \(\ds 2 \times 1201\) | ||||||||||||
\(\ds 4801\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 4802\) | \(=\) | \(\ds 2 \times 7^4\) | ||||||||||||
\(\ds 4803\) | \(=\) | \(\ds 3 \times 1601\) | ||||||||||||
\(\ds 7202\) | \(=\) | \(\ds 2 \times 13 \times 277\) | ||||||||||||
\(\ds 7203\) | \(=\) | \(\ds 3 \times 7^4\) | ||||||||||||
\(\ds 7204\) | \(=\) | \(\ds 2^2 \times 1801\) | ||||||||||||
\(\ds 9603\) | \(=\) | \(\ds 3^2 \times 11 \times 97\) | ||||||||||||
\(\ds 9604\) | \(=\) | \(\ds 4 \times 7^4\) | ||||||||||||
\(\ds 9605\) | \(=\) | \(\ds 5 \times 17 \times 113\) | ||||||||||||
\(\ds 12 \, 004\) | \(=\) | \(\ds 2^2 \times 3001\) | ||||||||||||
\(\ds 12 \, 005\) | \(=\) | \(\ds 5 \times 7^4\) | ||||||||||||
\(\ds 12 \, 006\) | \(=\) | \(\ds 2 \times 3^2 \times 23 \times 29\) | ||||||||||||
\(\ds 14 \, 405\) | \(=\) | \(\ds 5 \times 43 \times 67\) | ||||||||||||
\(\ds 14 \, 406\) | \(=\) | \(\ds 6 \times 7^4\) | ||||||||||||
\(\ds 14 \, 407\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 16 \, 806\) | \(=\) | \(\ds 2 \times 3 \times 2801\) | ||||||||||||
\(\ds 16 \, 807\) | \(=\) | \(\ds 7 \times 7^4\) | ||||||||||||
\(\ds 16 \, 808\) | \(=\) | \(\ds 2^3 \times 11 \times 191\) | ||||||||||||
\(\ds 19 \, 207\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 19 \, 208\) | \(=\) | \(\ds 8 \times 7^4\) | ||||||||||||
\(\ds 19 \, 209\) | \(=\) | \(\ds 3 \times 19 \times 337\) | ||||||||||||
\(\ds 21 \, 608\) | \(=\) | \(\ds 2^3 \times 37 \times 73\) | ||||||||||||
\(\ds 21 \, 609\) | \(=\) | \(\ds 9 \times 7^4\) | ||||||||||||
\(\ds 21 \, 610\) | \(=\) | \(\ds 2 \times 5 \times 2161\) | ||||||||||||
\(\ds 24 \, 009\) | \(=\) | \(\ds 3 \times 53 \times 151\) | ||||||||||||
\(\ds 24 \, 010\) | \(=\) | \(\ds 10 \times 7^4\) | ||||||||||||
\(\ds 24 \, 011\) | \(=\) | \(\ds 13 \times 1847\) | ||||||||||||
\(\ds 26 \, 410\) | \(=\) | \(\ds 2 \times 5 \times 19 \times 139\) | ||||||||||||
\(\ds 26 \, 411\) | \(=\) | \(\ds 11 \times 7^4\) | ||||||||||||
\(\ds 26 \, 412\) | \(=\) | \(\ds 2^2 \times 3 \times 31 \times 71\) | ||||||||||||
\(\ds 28 \, 811\) | \(=\) | \(\ds 47 \times 613\) | ||||||||||||
\(\ds 28 \, 812\) | \(=\) | \(\ds 12 \times 7^4\) | ||||||||||||
\(\ds 28 \, 813\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 31 \, 212\) | \(=\) | \(\ds 2^2 \times 3^3 \times 17^2\) | ||||||||||||
\(\ds 31 \, 213\) | \(=\) | \(\ds 13 \times 7^4\) | ||||||||||||
\(\ds 31 \, 214\) | \(=\) | \(\ds 2 \times 15 \, 607\) | ||||||||||||
\(\ds 33 \, 613\) | \(\text {is}\) | \(\ds \text {prime}\) | ||||||||||||
\(\ds 33 \, 614\) | \(=\) | \(\ds 14 \times 7^4\) | ||||||||||||
\(\ds 33 \, 615\) | \(=\) | \(\ds 415 \times 3^4\) | ||||||||||||
\(\ds 33 \, 616\) | \(=\) | \(\ds 2101 \times 2^4\) |
Hence the smallest valid triplet is $\tuple {33 \, 614, 33 \, 615, 33 \, 616}$.
$\Box$
Case $4$: the numbers are divisible by $2^4, 3^4, 5^4$ respectively
We utilise Chinese Remainder Theorem.
We are to solve the system of linear congruences:
- $x \equiv b_1 \pmod {2^4}$
- $x \equiv b_2 \pmod {3^4}$
- $x \equiv b_3 \pmod {5^4}$
where $\set {b_1, b_2, b_3} = \set {1, 0, -1}$.
First note the linear congruences:
- $3^4 5^4 x \equiv 1 \pmod {2^4}$
- $2^4 5^4 x \equiv 1 \pmod {3^4}$
- $2^4 3^4 x \equiv 1 \pmod {5^4}$
have solutions $1, 46, 231$ respectively.
Thus our system of linear congruences has the solution:
\(\ds x_0\) | \(=\) | \(\ds 3^4 5^4 b_1 + 2^4 5^4 \times 46 b_2 + 2^4 3^4 \times 231 b_3\) | \(\ds \pmod {2^4 3^4 5^4}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 50 \, 625 b_1 + 460 \, 000 b_2 + 299 \, 376 b_3\) | \(\ds \pmod {810 \, 000}\) |
Now we assign $\set {b_1, b_2, b_3}$ to $\set {1, 0, -1}$.
The solutions are:
- $\tuple {0, 1, -1}: 160 \, 624$
- $\tuple {0, -1, 1}: -160 \, 624 \equiv 649 \, 376$
- $\tuple {-1, 0, 1}: 248 \, 751$
- $\tuple {1, 0, -1}: -248 \, 751 \equiv 561 \, 249$
- $\tuple {-1, 1, 0}: 409 \, 375$
- $\tuple {1, -1, 0}: -409 \, 375 \equiv 400 \, 625$
and none of these solutions are less than $33 \, 616$.
$\blacksquare$
Historical Note
This result is reported by David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$ as the work of Stephane Vandemergel, but details are lacking.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $33,614$