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I’m looking for positive integer solutions to $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime.

Background.

I was looking at “Primes which are the sum of three nonzero 8th powers” https://oeis.org/A283019 and the like, and wondered if there are rules similar to those for a prime being the sum of two squares.

My efforts. I’ve found primes of the form $x^n+y^n+z^n$ up to $n=19$.

For $n=1$ solutions are trivial, for $n=2$ they are abundant. There are plenty of results for $n=3$ and they’re not uncommon for $n=4$.

However, for $\color{blue}{n=5}$ I’ve found just two lonely solutions.

$$(n,x,y,z=u,v,w,p)$$

$$(2,3,4,4=1,2,6,41)$$ $$(2,3,5,5=1,3,7,59)$$ $$(2,3,5,7=1,1,9,83)$$

$$(3,1,5,5=2,3,6,251)$$ $$(3,4,6,9=1,2,10,1009)$$ $$(3,1,9,9=4,4,11,1459)$$

$$(4,9,16,16=8,13,18,137633)$$ $$(4,4,18,19=1,6,22,235553)$$ $$(4,8,16,21=6,13,22,264113)$$

$$(\color{blue}5,11,183,209=19,168,216,604015282243)$$ $$(\color{blue}5,481,782,788=321,772,808,622015202536001)$$

My question. Can anyone find more solutions for $n=5$, or $n>5$, or give any insights, please.

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  • $\begingroup$ Google "taxicab number". It isn't exactly what you are looking for, but it'll get you close. $\endgroup$
    – Χpẘ
    Apr 6, 2017 at 20:02
  • $\begingroup$ Thanks for your interest. Sorry, I can’t see how Taxicab Numbers will help as I want primes and $x^3+y^3=(x+y)(x^2-xy+y^2)$ $\endgroup$
    – Old Peter
    Apr 7, 2017 at 8:57

3 Answers 3

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Taxicab numbers can be generalised to any number of terms.

With credit to Duncan Moore, author of Generalised Taxicab Numbers and Cabtaxi Numbers where I found suitable solutions, I can contribute: $$809^6+1851^6+2443^6=1277^6+1491^6+2489^6=253089021060516507491$$ $$511^6+2945^6+3285^6=1339^6+2457^6+3449^6=1909058509267895080811$$

A web search throws up this PDF entitled Complexity of Finding Values of the Generalized Taxicab Number which concludes that the problem is "at least supposed to be NP-Hard".

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  • $\begingroup$ Thank you very much for an excellent answer; I do appreciate the effort involved. I have only know these equations as “equal sums of like powers”, (6,3,3) and (6.3.3) so the term “Generalised Taxicab numbers” is new to me. $\endgroup$
    – Old Peter
    Apr 8, 2017 at 12:40
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    $\begingroup$ @OldPeter: Also, as I suspected, it is the case that,$$809^2 + 1851^2 + 2443^2 = 1277^2 + 1491^2 + 2489^2$$ $$809^6 + 1851^6 + 2443^6 = 1277^6 + 1491^6 + 2489^6$$ and similarly for Nick's second solution. In fact, as observed by Tom Womack, Duncan Moore, and others, by looking at databases, more than $90$% are multi-grade for $k=2,6$. No one really knows why. $\endgroup$ Apr 8, 2017 at 14:24
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    $\begingroup$ @Tito Referring to Old Peter's answer listing 5th powers, we find a connection also between $k=1,5$. Higher power solutions seem very rare so establishing whether this might extend to $k=n, n+4$ is probably difficult. $\endgroup$
    – nickgard
    Apr 10, 2017 at 5:43
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    $\begingroup$ @NickG: There are actually infinite families for $k=1,5$ and $k=2,6$, and a possible infinite one for $k=3,7$ found by Choudry. None is known for $k=4,8$, but there are sporadic ones for $k=1,3,9$. $\endgroup$ Apr 10, 2017 at 5:48
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    $\begingroup$ @OldPeter: No, it needs one more term per side. So the pattern starting with taxicabs numbers is $$3,2,2\\4,2,2\\5,3,3\\6,3,3\\7,4,4\\8,4,4\\9,5,5$$ though no multi-grades are yet known for $k=8$. $\endgroup$ Apr 11, 2017 at 1:43
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I just have a look at the primes solutions on my website Equal Sums of Like Powers, and find a very interesting one for your question.

$$19+179+241=43+139+257=439$$ $$19^3+179^3+241^3=43^3+139^3+257^3=19739719$$

All the numbers, including $x=19,y=179,z=241,u=43,v=139,w=257,n=3,p=19739719,x+y+z=439$ are Primes!!!

BTW, I also find the following one $$23+59+149+157+211=17+79+113+191+199=599$$ $$23^2+59^2+149^2+157^2+211^2=17^2+79^2+113^2+191^2+199^2$$ $$23^3+59^3+149^3+157^3+211^3=17^3+79^3+113^3+191^3+199^3$$ $$23^5+59^5+149^5+157^5+211^5=17^5+79^5+113^5+191^5+199^5=587777330999$$ All the items $(23,59,149,157,211,17,79,113,191,199)$ and the sums of each side ($599$), the sums of 5th powers ($587777330999$) are primes.

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  • $\begingroup$ Why write number???? The numbers aren't understanding. It should display formulas. $\endgroup$
    – individ
    Dec 5, 2017 at 6:40
  • $\begingroup$ @Chen Shuwen Thank you so much for taking an interest in this problem. Although I had found this solution, I had not recognised it as interesting out of my $100,000$ approx. solutions for third powers. Also, $19739719=10^3+214^3+215^3=49^3+131^3+259^3=87^3+88^3+264^3$ $\endgroup$
    – Old Peter
    Dec 5, 2017 at 19:16
  • $\begingroup$ @Chen Shuwen I’m so confused! Your site gives just the one result for $(k=1,3)$ but, unless I’ve made a very silly error, there are many dozens of smaller solutions. For example, $13+43+47=19+31+53=103$ and $13^3+43^3+47^3=19^3+31^3+53^3=185527$ $\endgroup$
    – Old Peter
    Dec 6, 2017 at 19:48
  • $\begingroup$ @Old Peter Yes, you are right and the sample you give is wonderful. My site focus on finding all the possible type of $(k=k_1,k_2,...,k_n)$ with idea solutions, and the secrets behind these phenomena. The prime solution for $(k=1,3)$ is not difficult to found, by that for $(k=2,3)$ is not so easy. Would you try your method and provide one ? $\endgroup$ Dec 6, 2017 at 22:09
  • $\begingroup$ Do feel free to up-vote either, or both, my comment and question, if you found them helpful. It would be best to post $(k=2,3)$ as a new question, so others can also contribute. $\endgroup$
    – Old Peter
    Dec 7, 2017 at 15:31
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Using Duncan Moore’s solutions, I’ve found that solutions for $n=5$ are not as rare as I thought earlier. Here are a few of the smallest; many of these also have $x+y+z=u+v+w$.

$$80^5+219^5+270^5=132^5+154^5+283^5=1941923897099$$ $$317^5+1008^5+1052^5=413^5+642^5+1172^5=2332329237198157$$ $$195^5+537^5+1267^5=970^5+1007^5+1072^5=3309936251919439$$ $$125^5+395^5+1479^5=559^5+922^5+1448^5=7086510650848399$$ $$893^5+1039^5+1691^5=733^5+1231^5+1659^5=15605379807526343$$ $$519^5+911^5+1841^5=103^5+1431^5+1737^5=21813107171029351$$ $$167^5+1153^5+1843^5=483^5+809^5+1871^5=23300964032544043$$ $$58^5+1171^5+1920^5=198^5+1331^5+1890^5=28293760481131619$$ $$101^5+1092^5+1944^5=274^5+906^5+1957^5=29316755331623957$$ $$69^5+1559^5+1863^5=679^5+823^5+1989^5=31651525808548691$$ $$235^5+1669^5+1965^5=189^5+1817^5+1863^5=42247371948274349$$ $$1124^5+1798^5+2119^5=1223^5+1644^5+2174^5=63307290246821191$$ $$283^5+2042^5+2182^5=432^5+1744^5+2331^5=84968164403859307$$ $$491^5+1459^5+2407^5=332^5+1653^5+2372^5=87433928094323557$$ $$58^5+1828^5+2367^5=981^5+1259^5+2463^5=94712207910987743$$ $$1266^5+1678^5+2469^5=1117^5+1873^5+2423^5=108305315229854293$$ $$693^5+1693^5+2497^5=417^5+2213^5+2253^5=111140193938890643$$ $$997^5+1979^5+2641^5=1009^5+1962^5+2646^5=159821917701479857$$ $$877^5+1960^5+2666^5=770^5+2252^5+2541^5=164123524321248733$$ $$1438^5+1916^5+2777^5=1132^5+2478^5+2521^5=197120435154165401$$ $$1277^5+2208^5+2784^5=1557^5+1849^5+2863^5=223118389276174349$$ $$942^5+2461^5+2808^5=413^5+1407^5+3041^5=265591178424588301$$ $$1912^5+1946^5+3083^5=804^5+2646^5+2891^5=331987128193291451$$ $$969^5+2791^5+2881^5=1341^5+2129^5+3171^5=368689558572449201$$ $$1799^5+2801^5+3543^5=2143^5+2371^5+3629^5=749539223719548943$$ $$559^5+2869^5+3671^5=1231^5+2039^5+3829^5=861122005375239499$$

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