Problems & Puzzles: Puzzles

Puzzle 563.- P=a^2+b^3

JM Bergot sent the following puzzle:

One notices that 17 is the sum of a square and a cube in two
different ways: 17=1^3 + 4^2 = 2^3 + 3^2.

This is also true for 89=2^3 + 9^2 = 5^2 + 4^3.

Q1. Are there other primes like this?
Q2. Is two ways the maximum?

Contributions came from J. K. Andersen, Fardi Lian, Torbjörn Alm, Luis Rodríguez, Seiji Tomita, Jeff Heleen, W. E. Clark, E. Vantiaghem & Fred Schalekamp.

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Andersen wrote:

Primes that can be written as a sum of a positive square and a positive cube in more than one way.
17, 89, 233, 449, 577, 593, 1289, 1367, 1601, 1753, 2089, 2521, ...

http://oeis.org/A173795 by Donovan Johnson:
Smallest prime that is the sum of a square and a positive cube in n different ways.
3, 2, 17, 2089, 65537, 3193361, 445341529, 4190216689, 25140740257, 813368268793, 333413867957257.

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Farid wrote:

 way(n) first prime 0 3 1 2 2 17 3 2089 4 65537 5 3193361 6 445341529 7 4190216689 8 25140740257 9 813368268793 10 333413867957257

333413867957257 = 18202887^2 + 12742^3 = 18190720^2 + 13593^3 = 16205565^2 + 41368^3 = 15621373^2 + 44712^3 = 14905630^2 + 48093^3 = 12187395^2 + 56968^3 = 11330919^2 + 58966^3 = 10486383^2 + 60682^3 = 9216035^2 + 62868^3 = 3854589^2 + 68296^3.

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Alm wrote:

Run up to a,b = 20000.

I counted nunber of solutions with 3 or more combinations with the same result.

The program found 15116448 primes.

Number of three-way solutions:2274
Number of four-way solutions:121
Number of five-way solutions:32

First solutions:

2089 = 19^2+12^3 =33^2+10^3 = 45^2+4^3
65537 = 255^2+8^3 =219^2+26^3 = 256^2+1^3 = 122^2+37^3
3193361 = 1047^2+128^3 =1606^2+85^3 = 1481^2+100^3 = 285^2+146^3 = 1769^2+40^3

Largest  solutions:

12642057289 = 17767^2+2310^3 =11071^2+2322^3 = 3000^2+2329^3
863258777 = 19899^2+776^3 =16384^2+841^3 = 13738^2+877^3 = 8005^2+928^3
333290161 = 17063^2+348^3 =14535^2+496^3 = 15256^2+465^3 = 13881^2+520^3 = 6957^2+658^3

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Luis wrote:

For the first question:

It is very probable that there are infinitely many
solutions, because any number can be decomposed
as the difference of two squares.
Take any two even numbers W,  Z.  (Z > W).
Its difference must not be divisible by 3.
N = Z^3 - W^3 . Decompose N = X^2 - Y^2
Then  X^2 + W^3 = Y^2 + Z^3
This can easily  to be a prime because there are many
possibilties.

Examples:

Z        W        Z^3 - W^3      X        Y      X^2+ W^3    Y^2 +  Z^3

10        8          488              63      59      4481              4481

16      10        3096              61      25      4721             4721

18      12        4104              65      11      5953             5953

40      38        9128           177     149     86201         86201

38      18      33472           539     507   296353       296353

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Tomita wrote:

Q2.

1. 3 ways

2089 =  19^2 +   12^3
=  33^2 +   10^3
=  45^2 +    4^3

2. 4 ways

65537 = 122^2 +   37^3
= 219^2 +   26^3
= 255^2 +    8^3
= 256^2 +    1^3

3. 5 ways

3193361 =  285^2 +  146^3
= 1047^2 +  128^3
= 1481^2 +  100^3
= 1606^2 +   85^3
= 1769^2 +   40^3

4. 6 ways

445341529 =  2523^2 +  760^3
= 11195^2 +  684^3
= 20773^2 +  240^3
= 20898^2 +  205^3
= 20955^2 +  184^3
= 21023^2 +  150^3

Are there a solution of seven ways?

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Jeff wrote:

Q1. Primes <1000 that are the sum of a cube and a
square in 2 different ways: 17, 89, 233, 449, 577,
and 593.

Q2.
3 ways
2089 =  4^3 + 45^2
= 10^3 + 33^2
= 12^3 + 19^2

4 ways
65537 =  1^3 + 256^2
=  8^3 + 255^2
= 26^3 + 219^2
= 37^3 + 122^2

5 ways
3193361 =  40^3 + 1769^2
=  85^3 + 1606^2
= 100^3 + 1481^2
= 128^3 + 1047^2
= 146^3 +  285^2

6 ways
445341529 = 150^3 + 21023^2
= 184^3 + 20955^2
= 205^3 + 20898^2
= 240^3 + 20773^2
= 684^3 + 11195^2
= 760^3 +  2523^2

7 ways
4190216689 =   72^3 + 64729^2
=  454^3 + 64005^2
=  762^3 + 61219^2
= 1330^3 + 42867^2
= 1425^3 + 36008^2
= 1554^3 + 20915^2
= 1572^3 + 17479^2

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Clark wrote:

Using a little trick I learned from Robert Israel
to find taxicab numbers using Maple, I found the following
quickly. I will be interested to see if someone can do
better.

Best wishes,

--Edwin
-----------------------------------------------------------
There appear to be many cases where a prime is the
sum of a square and a cube in more than one way. My
best results are where there are primes that occur
as a square and a prime in 4 different ways:

65537 = 1^3 + 256^2 =
8^3 + 255^2 =
26^3 + 219^2 =
37^3 + 122^2;

93241 = 6^3 + 305^2 =
28^3 + 267^2 =
40^3 + 171^2 =
45^3 + 46^2;

191969 = 5^3 + 438^2 =
10^3 + 437^2 =
14^3 + 435^2 =
32^3 + 399^2;

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Fred wrote:

Yes, there are many solutions and bigger ones:

For two different ways I found the next solution (P,A,B):  (89,9,2) and (89,5,4)
Two is not the maximum. up to P=2800000 I found:

Three ways: (4481,59,10) , (4481,63,8) , (4481,66,5)

Four ways: (65537,122,37) , (65537,219,26) , (65537,255,8) , (65537,256,1)

Greetings from a rainy Holland.

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Emmanuel wrote:

About Q1 : there are many primes that are twice the sum of a square and a cube.  The first ones are :

17,89,233,449,577,593,1289,1367,1601,1753,2521,3391,4721,5953,6121,6427,7577,8081,9649,... (not in the OEIS).

About Q2 : there are primes that are sum of a square and a cube in more than two ways.  The best I could find is the prime  3199961 with 5 such decompositions :

3193361 = 40^3+ 1769^2

3193361 = 85^3+ 1606^2

3193361 = 100^3+ 1481^2

3193361 = 128^3+ 1047^2

3193361 = 146^3+ 285^2

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On July 30, 2012, Kevin Acres wrote:

There's always this little gem with 14 solutions.

107122676734733201 =

460826^3 +  96236115^2
457750^3 + 105867851^2
447428^3 + 132481143^2
415601^3 + 187984920^2
403688^3 + 203312727^2
385333^3 + 223400642^2
344065^3 + 257666476^2
339826^3 + 260535965^2
319330^3 + 273056899^2
180440^3 + 318194601^2
128726^3 + 324021045^2
83413^3 + 326408198^2
56276^3 + 327023625^2
18076^3 + 327286985^2

11 & 12 are known as well. These are from a previous Noam Elkies search. See http://wstein.org/home/wstein/www/home/noam/j=0/10d. Here are the breakdowns for Noam's numbers:

4417190430889897 =
147192^3 + 35045747^2
141972^3 + 39441043^2
136516^3 + 43278099^2
131296^3 + 46409331^2
118938^3 + 52294015^2
85032^3 + 61663373^2
72477^3 + 63533258^2
69322^3 + 63906657^2
32146^3 + 66211569^2
28866^3 + 66280751^2
3082^3 + 66461727^2

84658174289284249 =
434424^3 + 51689915^2
409420^3 + 126606693^2
387910^3 + 162134943^2
386842^3 + 163610919^2
345880^3 + 208037307^2
332425^3 + 218913432^2
306582^3 + 236308609^2
230374^3 + 269131425^2
163977^3 + 283282696^2
138820^3 + 286326693^2
133017^3 + 286887844^2
50590^3 + 290738193^2

"These are all the lowest known prime solutions as opposed to the proven lowest."

***

On Set 21, Kevin Acres wrote again:

Noam Elkies has now performed an extensive search and has located the lowest primes p for which p is sum of a positive cube and a square in 11 and 12 different ways, respectively.

These are:

1057543811051633
= 7534^3 + 32513323^2
= 33184^3 + 31953127^2
= 46552^3 + 30929945^2
= 57377^3 + 29472900^2
= 69374^3 + 26901003^2
= 87989^3 + 19399158^2
= 94369^3 + 14735668^2
= 94874^3 + 14267997^2
= 95114^3 + 14038467^2
= 97952^3 + 10850535^2
= 101828^3 + 1302009^2

1448734752622601
= 30668^3 + 37681437^2
= 42326^3 + 37052775^2
= 49498^3 + 36434353^2
= 55000^3 + 35810051^2
= 68585^3 + 33557676^2
= 68890^3 + 33493199^2
= 78020^3 + 31206051^2
= 85838^3 + 28570377^2
= 88258^3 + 27590783^2
= 94820^3 + 24417699^2
= 105368^3 + 16700163^2
= 111901^3 + 6894130^2

The link that I previously gave to Noam's data is now inactive, you may like to remove it from the page.

Also please can you ensure that attribution is given to Noam for 1057543811051633 and 1448734752622601 and also for comfirming the status of my earlier 107122676734733201

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