The equation $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime.

Question

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.

Answer 1

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".

Answer 2

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 $x+y+z=u+v+w$.

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

Answer 3

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.