Bus Numbers

Find numbers from $10 - 999$ such that the number can be given in the form $x^4 + y^4$ in $n$ distinct ways where $x, y$ are all positive integers.

Make a list.

These are the bus numbers (Reason $1$ , those numbers are all bus routes. Reason $2$ , I love buses.)

All integers are positive.

#NumberTheory

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Do you mean sum of digits of the number?

Aryan Sanghi - 11 months, 1 week ago

@Yajat Shamji, since the freedom is there for $n$ to be positive and negative, any number can be achieved, right? Have I misunderstood the question, because it looks trivial then

Mahdi Raza - 11 months, 1 week ago

I guess maybe $n$ is a constant, so we have to decide considering it to be constant.

Aryan Sanghi - 11 months, 1 week ago

I agree with @Mahdi Raza. We can choose $n$ , so any number, not even till $999$ can be made.

Vinayak Srivastava - 11 months, 1 week ago

@Mahdi Raza, @Vinayak Srivastava, @Aryan Sanghi - I have changed the conditions - check again.

If nobody gives the full list before Friday, I'll post the first number.

Yajat Shamji - 11 months, 1 week ago

@Mahdi Raza, you know the definition of the Hardy-Ramanujan taxicab numbers, right?

All I did is change the exponent from $3$ to $4$ and named it bus numbers.

Yajat Shamji - 11 months, 1 week ago

Clue $1$ of $2$ : Here is the definition of the Hardy-Ramanujan taxicab number:

In mathematics, the $n$ th taxicab number, typically denoted Ta( $n$ ) or Taxicab( $n$ ), also called the $n$ th Hardy–Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in $n$ distinct ways. The most famous taxicab number is $1729 = \text{Ta}(2) = 1^3 + 12^3 = 9^3 + 10^3$ .

@Mahdi Raza, @Aryan Sanghi, @Vinayak Srivastava

Yajat Shamji - 11 months, 1 week ago

Here it is

Code

b = []

for i in range(0, 10000000):
    b.append(0)

for x in range(1, 32):
    for y in range(1, 32):
        b[x * x * x * x + y * y * y * y] += 1

for i in range(1, 26):
    print("For n = %s:" % (i), end = ' ')
    for j in range(1, 1000000):
        if b[j] == i:
            print(j, end = ' ')
    print()

Output

For n = 1: 2 32 162 512 1250 2592 4802 8192 13122 20000 29282 41472 57122 76832 101250 131072 167042 209952 260642 320000 388962 468512 559682 663552 781250 913952 
For n = 2: 17 82 97 257 272 337 626 641 706 881 1297 1312 1377 1552 1921 2402 2417 2482 2657 3026 3697 4097 4112 4177 4352 4721 5392 6497 6562 6577 6642 6817 7186 7857 8962 10001 10016 10081 10256 10625 10657 11296 12401 14096 14642 14657 14722 14897 15266 15937 16561 17042 18737 20737 20752 20817 20992 21202 21361 22032 23137 24641 24832 27297 28562 28577 28642 28817 29186 29857 30736 30962 32657 35122 35377 38417 38432 38497 38561 38672 39041 39712 40817 42512 43202 44977 48416 49297 50626 50641 50706 50881 51250 51921 53026 53057 54721 57186 59152 60625 65266 65537 65552 65617 65792 66161 66832 66977 67937 69632 71361 72097 75536 79186 80177 83522 83537 83602 83777 84146 84817 85922 86272 87617 89041 90082 93521 94097 98162 103952 104257 104977 104992 105057 105232 105601 106272 107377 109072 111537 112082 114976 116161 119617 121937 125712 130322 130337 130402 130577 130946 131617 132722 133537 134146 134417 136882 140321 143392 144962 149057 151057 155601 158882 160001 160016 160081 160256 160625 161296 162401 164096 166561 168737 170000 170512 174641 180736 180946 188497 188561 194482 194497 194562 194737 195106 195777 195857 196882 198416 198577 201042 204481 209122 210625 213842 215217 223042 225536 232897 234257 234272 234337 234512 234881 235297 235552 236657 238352 240817 243521 244256 245106 248897 254992 260017 262817 264976 272672 278002 279842 279857 279922 280097 280466 281137 282242 283937 284881 286402 289841 290321 294482 299457 299792 300577 308402 317777 318257 324802 330466 331777 331792 331857 332032 332401 333072 334177 335872 338337 339232 341776 345377 346417 352512 354481 360337 363362 364577 370192 382401 384817 390626 390641 390706 390881 391250 391921 393026 394256 394721 397186 397312 400625 405266 410162 411361 415297 419186 428737 429041 436752 439841 441250 456161 456977 456992 457057 457232 457601 458272 459377 461072 462097 463537 466976 471617 474146 474322 477712 485537 491776 495392 495601 507601 514097 520946 522512 526257 531442 531457 531522 531697 532066 532737 533842 535537 538002 540497 541441 546082 550625 552177 560002 561952 566032 569857 582066 585106 587297 596977 611617 614657 614672 614737 614912 614962 615281 615952 616976 617057 618752 621217 624656 624881 629297 635392 636417 643217 651457 653072 661762 665281 670466 680192 691232 691441 698177 707282 707297 707362 707537 707906 708577 709682 711377 713842 717281 719632 721922 722401 725922 728017 735842 736817 744977 745697 757906 765697 772817 774656 788752 790802 809137 810001 810016 810081 810256 810625 811282 811296 812257 812401 814096 816561 820000 824641 830736 837602 838561 847601 848416 848912 860625 863217 867281 875536 893521 894497 901762 914976 922066 923522 923537 923602 923777 924146 924817 925922 927617 930082 933521 938162 940321 941537 944257 946432 952082 961937 970000 974146 987122 988417 989057 
For n = 3: 
For n = 4: 
For n = 5: 
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For n = 7: 
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For n = 10: 
For n = 11: 
For n = 12: 
For n = 13: 
For n = 14: 
For n = 15: 
For n = 16: 
For n = 17: 
For n = 18: 
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For n = 22: 
For n = 23: 
For n = 24: 
For n = 25:

Aryan Sanghi - 11 months, 1 week ago

@Yajat Shamji

Aryan Sanghi - 11 months, 1 week ago

I request you to show proof (i.e. show all the $n$ distinct ways for their respective number where $a = x^4 + y^4$ .) since nobody else has replied and I need it to strengthen your list because after all, I need proof before announcing that you have found the list. (as well as keeping you busy - that's the main reason overall.)

Also, a challenge: try and do the numbers from $10 - 9999$ after giving the proof for your list above.

And if you've finished, show proof for that list.

Then, I will ask you to find the limit for the $n$ distinct ways where $a = x^4 + y^4$ .

After that, you're free.

P.S. If you want after the limit question, I've got a super-challenge: find the ratio of the numbers that satisifes the condition $a = x^4 + y^4$ in $n$ distinct ways to the number ( $n$ ) distinct ways.

P.S.S. $a$ is the number which fulfills the condition $x^4 + y^4$ in $n$ distinct ways.

P.S.S.S. You don't have to anything after finding the list for $10 - 9999$ and giving the proof for that list. It's just I want to keep you busy. If you don't want to do anything after finding the list for $10 - 9999$ and giving the proof for that list, then ask.

@Aryan Sanghi

Yajat Shamji - 11 months, 1 week ago

See now @Yajat Shamji, it's the list till $10^6$ . I have few points

Brute force checks all combinations, so if says it a number, it is a number
You can't find ratio as no number will go above $n = 2$
I am not free as I have lots of studies for my JEE, so please stop saying I am free.

If anything was offensive, I am sorry for it.

Aryan Sanghi - 11 months, 1 week ago

@Aryan Sanghi – What about the proof for the list?

Yajat Shamji - 11 months, 1 week ago

@Aryan Sanghi – I mean, I still need it.

Yajat Shamji - 11 months, 1 week ago

@Aryan Sanghi – Because once you've given the proof, I can tell Mahdi and Vinayak.

Yajat Shamji - 11 months, 1 week ago

One of $x$ and $y$ must be less than $\sqrt[4] {1000}$ , I. e., must be in the range $[2,5]$ ,while the other in the range $[1,4]$ . If by $n$ distinct ways we mean the pairs $(x_i,y_i)$ and $(y_i,x_i)$ are not distinct (as is the case with taxicab numbers), then there is no solution in that range. In fact, the smallest such number is a nine digit number $635318657$ , which can be expressed as the sum of the fourth powers of two integers in two different ways : $635318657=59^4+158^4=133^4+134^4$ .

A Former Brilliant Member - 11 months, 1 week ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$