Problems & Puzzles: Puzzles

Puzzle 1088 p = x U y such that p = x^a + y^b Paolo Lava sent the following nice puzzle. I am looking for the primes p = x U y (I mean concatenation with U) such that p = x^a + y^b, for some a and b. 43 = 4^2 + 3^3 89 = 8^1 + 9^2 809 = 80^1 + 9^3 65609 = 6560^1 + 9^5 496063 = 496^2 + 63^3 590489 = 59048^1 + 9^6 5882353 = 588^2 + 2353^2 47829689 = 4782968^1 +9^8 I noted that A239719 contains the primes p with y=9: 89, 809, 65609, 590489, 47829689 . In fact other terms are: 34867844009 = 3486784400^1 + 9^11 313810596089 = 31381059608^1 + 9^12 25418658283289 = 2541865828328^1 + 9^14 Q Other values?

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During the week ending on June 3, 2022, contributions came from Paul Cleary, Giorgos Kalogeropoulos, Jean-Marc Rebert

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Paul wrote:

Using the pattern of x^2 + y^2 there is one more solution

 

101 = 10^2 + 1^2

 

I searched up to a concatenated length of 10^80 with no more solutions.

 

Using the pattern of x^2 + y^3 there are other  solutions

 

101 = 10^2 + 1^3

9309600863 = 93096^2 + 863^3

 

Using the pattern of x^1 + y^b

 

13508517176729920889 = 1350851717672992088 ^1 + 9^20

581497370030400596903901689 = 58149737003040059690390168 ^1 + 9^28

47101286972462448349216036889 = 4710128697246244834921603688 ^1 + 9^30

119725151825620197886027400267170471056809 = 11972515182562019788602740026717047105680 ^1 + 9^43

46383976865881019793281501678905914543189676980089 = 4638397686588101979328150167890591454318967698008 ^1 + 9^52

62657874821779703792562241943419303322066944468106652748595980508009 = 62657874821779703792562241943419303322066
94446810665274859598050800 ^1 + 9^71

750172357610828013477547996399856313028435557734374670493358268049602018628424854624175111767958862537004809 =
75017235761082801347754799639985631302843555773437467049335826804960201862842485462417511176795886253700480 ^1 + 9^113

 

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Giorgios wrote:

Some other values are:

998000999 = 998000 + 999^3
 

2058911320946489 = 205891132094648 + 9^16
 

13508517176729920889 = 1350851717672992088 + 9^20
 

Also, if leading zeros are allowed, we can also have

9309600863 = 93096^2 + 00863^3 

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Jean-Marc wrote:

1) solutions with a=1 and y=9

10x+9 = x^1  + 9^b
iff 9x = 9*(9^(b-1) - 1)
iff x = 9^(b-1)-1

for b < 1000, I found the following primes p :

b, p = x^a + y^b

2 89 = 8^1 + 9^2 is prime
3 809 = 80^1 + 9^3 is prime
5 65609 = 6560^1 + 9^5 is prime
6 590489 = 59048^1 + 9^6 is prime
8 47829689 = 4782968^1 + 9^8 is prime
11 34867844009 = 3486784400^1 + 9^11 is prime
12 313810596089 = 31381059608^1 + 9^12 is prime
14 25418658283289 = 2541865828328^1 + 9^14 is prime
16 2058911320946489 = 205891132094648^1 + 9^16 is prime
20 13508517176729920889 = 1350851717672992088^1 + 9^20 is prime
28 581497370030400596903901689 = 58149737003040059690390168^1 + 9^28 is prime
30 47101286972462448349216036889 = 4710128697246244834921603688^1 + 9^30 is prime
36 25031555049932416013155719860858489 = 2503155504993241601315571986085848^1 + 9^36 is prime
43 119725151825620197886027400267170471056809 = 11972515182562019788602740026717047105680^1 + 9^43 is prime
52 46383976865881019793281501678905914543189676980089 = 4638397686588101979328150167890591454318967
698008^1 + 9^52 is prime
71 62657874821779703792562241943419303322066944468106652748595980508009 = 6265787482177970379256224
194341930332206694446810665274859598050800^1 + 9^71 is prime
113 75017235761082801347754799639985631302843555773437467049335826804960201862842485462417511176795
8862537004809 = 75017235761082801347754799639985631302843555773437467049335826804960201862842485462
417511176795886253700480^1 + 9^113 is prime
165 31316379552759662951457854619960196737773512607904438893704116750444100059230707803005931919727
511216375940393866253169135877932893109733818390104411143809609 = 313163795527596629514578546199601
96737773512607904438893704116750444100059230707803005931919727511216375940393866253169135877932893
10973381839010441114380960^1 + 9^165 is prime
180 6447764839223404737303268076687569183207508331491242266475115507659298210229061105228251588514
338866152628514376415103931020365622724108566486964210985397367805728955776889 = 64477648392234047
3730326807668756918320750833149124226647511550765929821022906110522825158851433886615262851437641
510393102036562272410856648696421098539736780572895577688^1 + 9^180 is prime
181 580298835530106426357294126901881226488675749834211803982760395689336838920615499470542642966
29049795373656629387735935379183290604516977098382677898868576310251560601992009 = 580298835530106
4263572941269018812264886757498342118039827603956893368389206154994705426429662904979537365662938
773593537918329060451697709838267789886857631025156060199200^1 + 9^181 is prime
243 8446704174255209088372902335474824927553340247681495424551857464957807167608398206264887925605
1555665083479186566623403796205363741905141028029593512040152326635450927487934447144865776340579
36717779356896928938844844508828010336809 = 844670417425520908837290233547482492755334024768149542
4551857464957807167608398206264887925605155566508347918656662340379620536374190514102802959351204
01523266354509274879344471448657763405793671777935689692893884484450882801033680^1 + 9^243 is prime
455 1683058786555803077915589244685555004301622555148887667073337073576423501352177252153078546931
46758586512919731967610486067262033824174446574496652807773860608806637589873185403502906035004509
09362713039358364458971715221807738329271245078773290419548043024717323828141430896180801611168825
12823121543402210538908312182245986967680717499026613833532355525986872275622455366239105438917295
31889899152671829652031745134263254979305845609 = 168305878655580307791558924468555500430162255514
8887667073337073576423501352177252153078546931467585865129197319676104860672620338241744465744966
5280777386060880663758987318540350290603500450909362713039358364458971715221807738329271245078773
2904195480430247173238281414308961808016111688251282312154340221053890831218224598696768071749902
66138335323555259868722756224553662391054389172953188989915267182965203174513426325497930584560^1 + 9^455 is prime
622 3842362736388431959623448285400900159324661931340994149646069735844590686734708031756608333715043
7930896913394491132736586414080361672212371224617074162353776913062119187781422985101518636208754
7493594505880189923364196038752171620215470733568780293460447126507473337566918288418940331917448
2516666882387265180315427992134976254311311995903893513833152617495152574612692935167824627449834
6151705950857257794548372938827968402563964041415328503607829696049886633621834860825639465865672
4220814509745383974119158065871405676068994754714432984195591495789878126077405452786180730242721
222495352089 = 38423627363884319596234482854009001593246619313409941496460697358445906867347080317
56608333715043
7930896913394491132736586414080361672212371224617074162353776913062119187781422985101518636208754
7493594505880189923364196038752171620215470733568780293460447126507473337566918288418940331917448
2516666882387265180315427992134976254311311995903893513833152617495152574612692935167824627449834
6151705950857257794548372938827968402563964041415328503607829696049886633621834860825639465865672
4220814509745383974119158065871405676068994754714432984195591495789878126077405452786180730242721
22249535208^1 + 9^622 is prime

2) solutions with a=1 and y=10^c-1 and c > 0

x*10^c + 10^c-1 = x^1 + (10^c-1)^b iff

x(10^c-1) = (10^c-1)((10^c-1)^(b-1)-1) iff

x = (10^c-1)^(b-1) - 1

for b < 1000, I found the following primes p :

b, p = x^a + y^b

For b < 100 and 0 < c < 10 :

c=1

b, p = x^a + y^b
2 89 = 8^1 + 9^2 is prime
3 809 = 80^1 + 9^3 is prime
5 65609 = 6560^1 + 9^5 is prime
6 590489 = 59048^1 + 9^6 is prime
8 47829689 = 4782968^1 + 9^8 is prime

c=3

b, p = x^a + y^b
3 998000999 = 998000^1 + 999^3 is prime
8 993020965034979006998999 = 993020965034979006998^1 + 999^8 is prime

c=4

b, p = x^a + y^b
2 999998999999 = 999998^1 + 999999^2 is prime

c=7

b, p = x^a + y^b
5 99999960000005999999600000009999999 = 9999996000000599999960000000^1 + 9999999^5 is prime

c=9

b, p = x^a + y^b
2 999999998999999999 = 999999998^1 + 999999999^2 is prime

Pour b < 100 et c<=10 :

c = 2
b, n = x^a  + y^b
26 7778213593991467720087394915620714343003729700249899 = 77782135939914677200873949156207143430
037297002498^1 + (99)^26 is prime
36 703447694999569431287210146870025849536948217143317263208109053905349899 = 703447694999569431
2872101468700258495369482171433172632081090539053498^1 + (99)^36 is prime
72 48989027300420518726426892791225794310134638561760794471963173800654467315495964324732531641
3238815828815023796054051810346958160298363015709899 = 48989027300420518726426892791225794310134
63856176079447196317380065446731549596432473253164132388158288150237960540518103469581602983630
157098^1 + (99)^72 is prime

c = 3
b, p = x^a  + y^b
3 998000999 = 998000^1 + (999)^3 is prime
8 993020965034979006998999 = 993020965034979006998^1 + (999)^8 is prime

c = 4
b, p = x^a  + y^b
66 9935207563876214836493113858667425532690025872467497964981669334619452594233520538623819427599
0727949655358500417871842259604067105833102717740911206887935930563995437087406660910224929384309
1777298656034318202295579729651488058230019459820296436797920006499989999 = 9935207563876214836493
1138586674255326900258724674979649816693346194525942335205386238194275990727949655358500417871842
2596040671058331027177409112068879359305639954370874066609102249293843091777298656034318202295579
72965148805823001945982029643679792000649998^1 + (9999)^66 is prime

c = 6
b, p = x^a  + y^b
2 999998999999 = 999998^1 + (999999)^2 is prime

c = 7
b, p = x^a  + y^b
5 99999960000005999999600000009999999 = 9999996000000599999960000000^1 + (9999999)^5 is prime
68 99999330002210995209507664790342361979560903524442357284232123945521417254519974031263045681
70840214998598220137042043528124834771273501001312688679881835818471035799954488175511778982805
56455543197898264482785480940986261091651256273805472709833721519746044498715563256164676262871
57804333337374316213123442222811973958507301062511722091662573690449922167610926446250696288198
37851018332037653236277702907763852696481980204304965764792335200047904999778900000669999998999
9999 = 99999330002210995209507664790342361979560903524442357284232123945521417254519974031263045
68170840214998598220137042043528124834771273501001312688679881835818471035799954488175511778982
80556455543197898264482785480940986261091651256273805472709833721519746044498715563256164676262
87157804333337374316213123442222811973958507301062511722091662573690449922167610926446250696288
1983785101833203765323627770290776385269648198020430496576479233520004790499977890000066999999
8^1 + (9999999)^68 is prime

c = 9
b, p = x^a  + y^b
2 999999998999999999 = 999999998^1 + (999999999)^2 is prime
18 99999998300000013599999932000000237999999381200001237599998055200002430999997569000001944799
9987624000006187999997620000000679999999864000000016999999998999999999 = 99999998300000013599999
93200000023799999938120000123759999805520000243099999756900000194479999876240000061879999976200
00000679999999864000000016999999998^1 + (999999999)^18 is prime

c = 10
b, p = x^a  + y^b
54 9999999947000000137799999765740000292824999713031500229574799845856920088632270955683864519499
0996123776246966678313562586070337703979903574965225056445759052041217146926756570716676421166855
0084042363558218197573350573957504861655311899306383818929497127267090287266337616183012646880726
4042495455074266463718644180781949915758682357897088243428416087828580505424092190034775067960200
9588413929664433216864297622375305805009003804431613549911367729001541430799977042520000286968499
9970717500000234259999998622000000005299999999989999999999 =  000029282499971303150022957479984585
6920088632270955683864519499099612377624696667831356258607033
7703979903574965225056445759052041217146926756570716676421166855008404236355821819757335057395750
4861655311899306383818929497127267090287266337616183012646880726404249545507426646371864418078194
9915758682357897088243428416087828580505424092190034775067960200958841392966443321686429762237530
580500900380443161354991136772900154143079997704252000028696849999707175000002342599999986220000
0000529999999998^1 + (9999999999)^54 is prime


3) solutions with y!=9 :

p = x^a + y^b

43 = 4^2 + 3^3 is prime
5882353 = 588^2 + 2353^2 is prime
998000999 = 998000^1 + 999^3 is prime


4) There are also the following solutions with x = 10^n (n>1)

examples :
101 = 10^2 + 1
103 = 10^2 + 3
107 = 10^2 + 7^1
109 = 10^2 + 9^1

... (many more up to)

109937 = 10^5 + 9937^1
109943 = 10^5 + 9943^1
109961 = 10^5 + 9961^1
109987 = 10^5 + 9987^1

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