More x^y y^x fun.



#15

How many integer values of x and y can you find that satisfy xy + yx = p, where p is prime and x > 1 and y > 1, can you find with your favorite calculator? I'll give you the first pair: 2 and 3.

Update: I added that x and y must both be greater than one because it would be too easy to use any number one less than a prime paired with a one, e.g. 10^1 + 1^10 = 11.

Edited: 14 Jan 2009, 4:09 a.m. after one or more responses were posted


#16

The firs pair must surely be 1 and 2?


#17

Yes, you are correct. I had primes on the mind and started with 2 and 3, but I was right in doing so, see the above changes.

Edited: 14 Jan 2009, 4:05 a.m. after one or more responses were posted


#18

Quote:
I had primes on the mind and started with 2 and 3.

IMHO 1 is a prime.

#19

"a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself."

I think the operative words are two and distinct. Of course you did say, IMHO. IMHO, opinions are neither right or wrong.

Edited: 14 Jan 2009, 4:09 a.m.


#20

"The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own."

Source http://mathworld.wolfram.com/PrimeNumber.html

#21

This is a fun problem...

[2,9] is the next pair, then [2,15] and [2,21]. After that we're talking pretty big numbers for my 32sii to digest.

Off the calc, I found [2,33]. [2,27] is not prime so [x,3+6n] is not always a solution. Still, I'd guess that there are an infinite number of solutions with 2 as one of the numbers. Are there other solutions that don't have 2 as one of the numbers?


note: After a little internet searching I found a whole lot more on this. There are indeed pairs that don't involve 2 as one of the numbers, but you're going to need something a whole lot faster than a 32sii to find them.

Edited: 14 Jan 2009, 6:35 a.m.


#22

Quote:
Are there other solutions that don't have 2 as one of the numbers?

Yes, the smallest prime being 24^5+5^24=59604644783353249.

Quote:
note: After a little internet searching I found a whole lot more on this. There are indeed pairs that don't involve 2 as one of the numbers, but you're going to need something a whole lot faster than a 32sii to find them.

Before I went to sleep last night I wrote a small brute-force search in UserRPL on my 50g to search all unique pairs with exponents < 100 (COMB(99,2) total pairs). I found all 16 pairs in a little over 6.5 hours. I expect that a C version at 75MHz to find them in about one minute.

I started on a 41cx/42s version, but need something smarter, not harder.


#23

It is hard to look for (x,y) pairs which verify the x^y+y^x=prim condition using only a HP, even a powerfull one !

I only get a list of suspects, but I have not found a time revelant way to pruve it :

List of Suspected pairs:
... ( 3,56)(18,19)(4,35)(16,17)(8,21)

List of Confirmed pairs :
( 5,24)( 2,33)( 2,21)( 2,15)( 2, 9)( 2, 3)

Rejected:
( 9,26)( 6,35)( 2,69)( 2,75)( 3,38)( 3,32) ...

Has anyone a convenient strategy to resolve this on an HP calculator ?

Edited: 15 Jan 2009, 1:44 p.m.


#24

Quote:
It is hard to look for (x,y) pairs which verify the x^y+y^x=prim condition using only a HP, even a powerfull one !

I had no problem finding the first 16, but it took 6.5 hours on the 50g. I'll write a C version this weekend and it will take about 60 seconds.
Quote:
List of Suspected pairs:
... ( 3,56)(18,19)(4,35)(16,17)(8,21)

Prime, Composite, Composite, Composite, Composite, Composite.

Good guesses however, all the pairs have opposite parity (odd/even), and all pairs are coprime. Clearly both conditions have to be met to be a candidate.

Quote:
List of Confirmed pairs :
( 5,24)( 2,33)( 2,21)( 2,15)( 2, 9)( 2, 3)

The last 5 are the only one that can be discovered on a 10-12 digit HP without the aid of some type of multiple precision library. Fortunately the 50g has multiple precision integers built-in. If you want a stab at the 71B try Valentin's MP Library for the 71B (http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv018.cgi?read=135156). Or, for the greatest of challenges, go for Peter's MP library for the 41 (http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv018.cgi?read=143386)!
Quote:
Rejected:
( 9,26)( 6,35)( 2,69)( 2,75)( 3,38)( 3,32) ...

I like your thinking. 4 out of the 6 2nd digits are correct.
Quote:
Has anyone a convenient strategy to resolve this on an HP calculator ?

There are COMB(n,2) pairs to check with n being the greatest exponent. You can cut that down by checking that each pair has opposite parity and is coprime. I am working on a 42S version that does this. I'll try to post this weekend. But I only expect to find 5 because of the lack of MP for the 42S.


Edited: 16 Jan 2009, 5:32 a.m.


#25

Thank you Egan for the links where I found a lot of very interresting and instructive stuff.

I was suspecting that my solver was not optimum, I now undoubtedly know that it is not efficient. The good thing by reading your posts I also have new idea and alternative way to try, if I can get the time...

Time is the keypoint in this problem. Due to lack of time and too slow "isprim" algorithms, I still stuck in proving the last few (x,y) pairs candidate give a prim x^y+y^x :

Searching for (x,y) is not so long especially considering the following restriction :

-- x<>y: Obviusly x=y lead to x^x+x^x=2.x^x which is composite (except (1,1) -> 2 the only odd prim which out of valid x>1 y>1 domain.

  • x<y : x and y are symetric (x,y) is equivalent to (y,x). To restrict domain of sear, one may only consider (x,y) whith x<y
  • x and y of different parity since (even,even) or (odd,odd) result to an odd sum which may not be prim (except for the odd prim 2 out of range)
  • coprime(x,y) : At first, I get trouble demonstrating it ! X+Y=p prim implies that X and Y are coprime. It take me some effort to understand why X=x^y and Y=y^x lead to x and y coprime as well.
  • coprime(x,y) and coprime(y,z) : (x,yz) only acceptable when (x,z) are also coprime. If not, x^(yz)+(yz)^x=p can be factorized (at least one) by at common factor of x and y and p never be prim !

    But x,y,z all coprime each other (x,y) correct did not prouve (x,yz) or (xz,y) produce a prim !

    I still investigating there if there is a way to eliminate (x,yz) candidate base on common factor or a combination of it.[lf]
Using these restriction restrict greatly the search loop of (x,y).
The problem, and the main time consumption in my implementation, is for testing primality of (x,y) -> p = x^y+y^x. Especially when p is higher of the 10-12 digits capability of my calculator or when p is higher than the last prim of my prims data file (containing nearly 7 million of pre-compute prims).

Is 523347633027360537213687137. prim ?

Greatly suspected pairs list:
( 3, 56) -> 5233476 3302736053 7213687137. Is prim ?
( 18, 19) -> 8123 6269565324 8917890473. Is prim ?
( 4, 35) -> 11 8059162071 7412804049. Is prim ?
( 16, 17) -> 3 4380909705 5019694337. Is prim ?
( 5, 24) -> 5960464 4783353249. Is prim ?
After testing it, over days and nights, using the 7603553 firsts prims I know, I still not sure its primality !

Confirmed pair list:
( 2, 33) -> prim 8589935681.
( 2, 21) -> prim 2097593.
( 2, 15) -> prim 32993.
( 2, 9) -> prim 593.
( 2, 3) -> prim 17.

Rejected pairs:
( 9, 26) -> composite 64610 8188923210 2802611217. = 247720068 3390506243. x 2608219.
( 8, 21) -> composite 922337207 4677635169. = 24 5992359227. x 37494547.


#26

Quote:
Greatly suspected pairs list:
(  3,  56) ->   5233476 3302736053 7213687137.  Is prim ?
( 18, 19) -> 8123 6269565324 8917890473. Is prim ?
( 4, 35) -> 11 8059162071 7412804049. Is prim ?
( 16, 17) -> 3 4380909705 5019694337. Is prim ?
( 5, 24) -> 5960464 4783353249. Is prim ?
After testing it, over days and nights, using the 7603553 firsts prims I know, I still not sure its primality !

My Nspire isprime() function tells me that:

3^56+56^3 is prime
18^19+19^18 is not prime
4^35+35^4 is not prime
16^17+17^16 is not prime
5^24+24^5 is prime
#27

Quote:
x<y : x and y are symetric (x,y) is equivalent to (y,x). To restrict domain of sear, one may only consider (x,y) whith x<y

Yes, that is why there are only COMB(n,2) pairs to test, where n is the greatest exponent, e.g.:

(3,2), (4,2), (4,3), (5,2), (5,3), (5,4), ...

Quote:
x and y of different parity since (even,even) or (odd,odd) result to an odd sum which may not be prim (except for the odd prim 2 out of range)

To add to that, oddeven is odd and evenodd is even. And odd + even is odd. Primes > 2 are all odd. Instead of testing for this one could alter their inner loop to skip every other number starting with 2 or 3 depending on an odd or even base.
Quote:
coprime(x,y) : At first, I get trouble demonstrating it ! X+Y=p prim implies that X and Y are coprime. It take me some effort to understand why X=x^y and Y=y^x lead to x and y coprime as well.

Assume that a + b = prime. If a = cm and b = dm, i.e. a and b share a common factor, then a + b = cm + dm = m(c + d) != prime, i.e. the sum of any pair of numbers that share a common factor will also share the same common factor.

As for ab + ba, it is the same since a and all of its factors get multiplied b times and vv., i.e. cm just becomes cm*cm*cm..., each still has m as a factor as will the sum.

The quickest test for coprime is a GCD of 1.

Quote:
Is 523347633027360537213687137. prim ?

Yes. BTW, for no charge you can setup an account here: http://www.sagenb.org/, and then type factor(523347633027360537213687137) or is_prime(523347633027360537213687137), but the server is often loaded, so I would not try to factor anything very large.

Edited: 17 Jan 2009, 12:06 p.m.

#28

Quote:
How many integer values of x and y can you find that satisfy xy + yx = p, where p is prime and x > 1 and y > 1, can you find with your favorite calculator? I'll give you the first pair: 2 and 3.

As explained above each pair must have opposite parity (odd/even) and be co-prime. This can reduce the number of tests by more than half.

All the programs below use the same basic algorithm:

Loop from 3 to n (largest base)
Loop from 2 or 3 depending on if outer loop is odd/even and skip every other number (i.e. opposite parity), end at outer loop - 1
If GCD = 1 (i.e. co-prime) then check for prime and print if prime

50g version. The 49g+/50g are the only calculators that I have with multiple precision integers.

%%HP: T(3)A(R)F(.);
\<< \-> n
\<<
3 n FOR i
3 i 2 MOD - i 1 - FOR j
i j GCD 1 == IF THEN
i j ^ j i ^ + DUP ISPRIME? IF THEN
i j ROT 3 \->LIST
ELSE
DROP
END
END
2 STEP
NEXT
\>>
\>>
Using 100 for n and timing it I got the following output:
17: { 3 2 17 }
16: { 9 2 593 }
15: { 15 2 32993 }
14: { 21 2 2097593 }
13: { 24 5 59604644783353249 }
12: { 32 15 43143988327398957279342419750374600193 }
11: { 33 2 8589935681 }
10: { 38 33 5052785737795758503064406447721934417290878968063369478337 }
9: { 54 7 4318114567396436564035293097707729426477458833 }
8: { 56 3 523347633027360537213687137 }
7: { 68 21 814539297859635326656252304265822609649892589675472598580095801187688932052096060144958129 }
6: { 69 8 205688069665150755269371147819668813122841983204711281293004769 }
5: { 75 34 7259701736680389461922586102342375953169154793471358981661239413987142371528493467259545421437269088935158394128249 }
4: { 76 9 3329896365316142756322307042065269797678257903507506764421250291562312417 }
3: { 81 80 14612087592038378751152858509524512533536096028044190178822935218486730194880516808459166772134240378240755073828170296740373082348622309614668344831750401 }
2: { 87 56 123696767198741648186287940563721003128015158572134981161748692560225922426827257262789498753729852662122870454448694253249972402126255218031127222474177 }
1: 23276.5845947

That last value is the number of seconds. Most of the time was spend checking for primarlity (checking the last number took over 1500 seconds).

My 2nd program was for the 42S. Without an MP library only 5 possible solutions can be found.

01  LBL "XYYX"      19  RCL "X"         37  GTO 03          55  GTO 01          
02 FIX 00 20 IP 38 Rv 56 LBL 03
03 1E3 21 2 39 XEQ "?PRI" 57 ISG "X"
04 / 22 MOD 40 1 58 GTO 00
05 3 23 - 41 X!=Y? 59 LBL 04
06 + 24 STO+ "Y" 42 GTO 02 60 RCL "X"
07 STO "X" 25 LBL 01 43 "(" 61 IP
08 LBL 00 26 RCL "X" 44 RCL "X" 62 ENTER
09 RCL "X" 27 IP 45 AIP 63 ENTER
10 1 28 RCL "Y" 46 |-"," 64 RCL "Y"
11 - 29 IP 47 RCL "Y" 65 IP
12 IP 30 XEQ "GCF" 48 AIP 66 Y^X
13 1E3 31 1 49 |-") = " 67 X<>Y
14 / 32 X!=Y? 50 XEQ 04 68 LASTX
15 2E-5 33 GTO 02 51 AIP 69 X<>Y
16 + 34 XEQ 04 52 PRA 70 Y^X
17 STO "Y" 35 1E12 53 LBL 02 71 +
18 3 36 X<=Y? 54 ISG "Y" 72 RTN
You'll need to provide your own GCF and ?PRI for computing and checking for GCD and primality. Versions that I used can be found here:
http://www.hpmuseum.org/software/42snumth.htm and here: http://www.hp42s.com/programs/prm/prm.html. Unfortunately, both primality testing programs failed to find 332 + 233 = 8589935681, which is prime. Both return 8589935681 as composite.

Since bases larger than 40 would exceed the largest integers on the 42S, an n of 40 was used with the following output sent to the printer:

(3,2) = 17
(9,2) = 593
(15,2) = 32993
(21,2) = 2097593
I did not have the time or desire to input this into my 42S and time it. All testing was done with Free42 and EMU42. Both run in about a second. EMU42 with Authentic Calculator Speed checked ran in about 5 minutes. Not bad at all. Again 332 + 233 = 8589935681, which is prime, was not identified.

Lastly, I wrote a 50g C program. Output (n = 250):

xyyx
----
max base = 250
1:(3,2) st:0 tt:0 p=17
2:(9,2) st:0 tt:0 p=593
3:(15,2) st:0 tt:0 p=32993
4:(21,2) st:0 tt:0 p=2097593
5:(24,5) st:0 tt:0 p=59604644783353249
6:(32,15) st:0 tt:0 p=43143988327398957279342419750374600193
7:(33,2) st:0 tt:0 p=8589935681
8:(38,33) st:0 tt:0 p=5052785737795758503064406447721934417290878968063369478337
9:(54,7) st:1 tt:1 p=4318114567396436564035293097707729426477458833
10:(56,3) st:0 tt:1 p=523347633027360537213687137
11:(68,21) st:1 tt:2 p=814539297859635326656252304265822609649892589675472598580095801187688932052096060144958129
12:(69,8) st:0 tt:2 p=205688069665150755269371147819668813122841983204711281293004769
13:(75,34) st:0 tt:2 p=7259701736680389461922586102342375953169154793471358981661239413987142371528493467259545421437269088935158394128249
14:(76,9) st:1 tt:3 p=3329896365316142756322307042065269797678257903507506764421250291562312417
15:(81,80) st:3 tt:6 p=14612087592038378751152858509524512533536096028044190178822935218486730194880516808459166772134240378240755073828170296740373082348622309614668344831750401
16:(87,56) st:1 tt:7 p=123696767198741648186287940563721003128015158572134981161748692560225922426827257262789498753729852662122870454448694253249972402126255218031127222474177
17:(114,67) st:17 tt:24 p=14877416035581437625382418693025659213718389161995860818124841388673684963203665153674781821433446993366770573625979847557897428218464508224911011186563057321746523584348117445155146293741592207500868288335433
18:(114,97) st:2 tt:26 p=31044002257937938068829512069328418720442849714746044259262070211475909156430718104897916371346903019988299736264997845256331029880527467485588680290321008690638402969435837489232709464321634204160348150282351066675254883643073
19:(122,9) st:14 tt:40 p=261568927457882874608733211757582315090892217214195250256575658313972901281170319830426649720495055337775965208077073
20:(135,32) st:34 tt:74 p=156764265941034957982331212844852467344711417043899710759469297619722251722129607859661177881884230709880082871203965476543290384119266534864181746784056675684904885421941056286488906715343079485633768193
21:(144,65) st:38 tt:112 p=1146894200299040723795729560557432803945230313807638769578093982012136321215160297940927699948343520671556845736219442653998944625369329903193883106139926361443140667237807076206116279779808347609272466625004356934372143216248343316093884886514556058622237058049
22:(158,45) st:44 tt:156 p=1612787603050274562069903478238630968344365069682295284380830826698222889674680197987083262414275400043686513974598153012882088977642138200701969570555398418510708201351413523014544364196803446185300014434337842494330504602516368646842185185091035058849770379993
23:(160,133) st:33 tt:189 p=6550320570424736017975912906823227230795756706291794704108177489835078496769415238689514208202469044620324030607457783482205821651727093725227403397073694702896425225342185751848441751740036285422160020451093004818459243817193874978308895037403004230124437546048120534386189574836751659857634392400828235697956677388632599317401261314620801
24:(171,98) st:59 tt:248 p=31597952610481735406184417800290574875942553425307879180826388419529262723040048711068602071994548107999174044084577517699846942953913418228435183384942999925853097569116224722566930758007858250860318446140839471731747637849343767087120159917109898890189838206328093476501956660894289333508728745376522299995609109005181880977444726036361913
25:(185,36) st:78 tt:326 p=824067225624450724350185185776877304119442769678571343023989650001883754551070680307504347916864451372704776163119070934582841099887572070617259829434205627038583354453077452767660757445997902514308790481190497037477812241451095936591256345061049919894048836391873825201358795351447541601
26:(206,51) st:299 tt:625 p=5747199543760759616874233229888257422940157808956554894989528303876688063774886847508851734925031076478824589152166195877212975767050907832290325148805207959265061065974594521872318497656340905582188769534415194253694453761740437565393308530950568782990088412137769599069158309021988800952585164712968388821327706584784875619399795456912191697392743257
27:(214,157) st:229 tt:854 p=83661493955934721940093572212231383493946119658337358148614804736256324921471160972435870342060927623728152367517269822843884194359726682507710240594072584001564534040253478907766066569333018912502014344554426181798194999272481888123729683855997762382503244058294295978132489987218780748626118191917375910685828051555739477358215225955442752598788025402065384853015539995876967147994467068334686232696554600733521322102525300704289117626991526368429784833137841843604953
28:(215,76) st:39 tt:893 p=237094969949224442749902986380791020043168977628753310230413588887866019807206687260728586774883158972331245704593379062623514507537767351876807280227464301767774824002902112830817761681285007322413139576110910042445342936924413085089951612853750811126254043830282762406659735949091917233649175898134000967364940467954696619147066269814574920691123415395618086403122820151726350048959757440141854187388001
29:(235,214) st:493 tt:1386 p=44385052412762688318951055173380319843487748708306634619440889745835408830738494165959460186649795039935869857701360175697514476212887207864294549983864003450722679260729153842868970919466290674158790879651861662496290527238948143838921484616959315821844518426079457762372930928602345374576342716989247843757891309322658042817360942693414966600088738885411412998933675452974393371268704164198882524739941965846960794214860883824710910263573766169516089092742362941226839480712862783430499156602698187227452508958045052406028997664027096899107636649
30:(237,200) st:66 tt:1452 p=220855883097298041197912187592864814478435487109452369765200775161577480905723392388044682757315234654167670063250350243744077256350845446337761180825366337002635606656007341132320168032281392575017521703513771018927360713671517013624864566435547143474670149962861625252760480437528208244008235645089927121069906768913003579993568524493534133751800322779056517412042477855290593538630256881723755480024801815845773154160553997782832989236043893126868182761752694180134718301605006478125705120066279786312075737518812303625846500724858615981588001
31:(248,87) st:308 tt:1760 p=10017852879331048206259365643153524162288395290235824439008838292948861954561026577732490631710064187459350801885314214526661356445274819838307275428757174304254585208499586166194622979711174028594438075409019427281714941478505682983882454321763489168841963884963368611530325032984768592374068808877912054221465743402174420737098853301865336967778491146415153029254969741768852576657071243601202773439328674914074447937954978745826356707638975566131088882293333821065981432759008193
Total time: 1902

The first 16 values also identivied by my 50g UserRPL program took 7 seconds with C @ 192MHz, vs. 23277 seconds with UserRPL @ 75MHz.

Universal HPGCC2, HPGCC3, and HPAPINE code. Put a real (not an integer on the stack--HPAPINE limitation). Press any key to interrupt and get intermediate results.

#include <hpgcc49.h>
#include <tommath.h>
#define OUTPUT printf("%s",buf); fprintf(fp,"%s",buf)
int gcd(int, int);
int main()
{
mp_int a, b;
mp_err ret;
int n, x, y, c=0, start, split, end, t;
FILE *fp;
char *filename="xyyx.txt", buf[5000];
    if((fp = fopen(filename,"w")) == NULL) {
char errormsg[30];
        sprintf(errormsg,"Cannot open: %s",filename);
#ifdef HPGCC2
sat_stack_push_string(errormsg);
sat_push_real(1);
#else
sat3_push_string(errormsg);
sat3_push_int_real(1);
#endif
return(0);
}
#ifdef HPGCC2
n = sat_pop_real();
sys_slowOff();
#else
n = sat3_pop_dbl(100);
cpu_setspeed(192*1000000);
#endif
    clear_screen();
mp_init(&a);
mp_init(&b);
sprintf(buf,"xyyx\n----\nmax base = %d\n\n",n); OUTPUT;
start = split = sys_RTC_seconds();
    for (x = 3; x <= n; x++) {
for (y = 3 - x % 2; y < x; y+=2) {
if(gcd(x,y) == 1) {
mp_set_int(&a, x);
mp_expt_d(&a, y, &a);
mp_set_int(&b, y);
mp_expt_d(&b, x, &b);
mp_add(&a, &b, &a);
mp_prime_is_prime(&a, 1, &ret);
if (ret) {
end = sys_RTC_seconds();
t = (end - split >= 0) ? end - split : end - split + 86400;
sprintf(buf,"%d:(%d,%d) st:%d ",++c,x,y,t); OUTPUT;
t = (end - start >= 0) ? end - start : end - start + 86400;
sprintf(buf,"tt:%d",t); OUTPUT;
fprintf(fp," p=");
mp_toradix(&a, buf, 10);
fprintf(fp,"%s",buf);
sprintf(buf,"\n"); OUTPUT;
split = end;
}
}
#ifndef HPAPINE
if(keyb_isAnyKeyPressed())
break;
#endif
}
#ifndef HPAPINE
if(keyb_isAnyKeyPressed())
break;
#endif
}
    end = sys_RTC_seconds();
t = (end - start >= 0) ? end - start : end - start + 86400;
sprintf(buf,"\nTotal time: %d\n",t); OUTPUT;
fclose(fp);
#ifdef HPGCC2
WAIT_CANCEL;
#else
SLOW_WAIT_CANCEL;
#endif
return (0);
}
int gcd(int x, int y)
{
int t;
    while (y) {
t = x;
x = y;
y = t % y;
}
return(x);
}

References:

http://www.leyland.vispa.com/numth/primes/xyyx.htm#results


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