A274026 Perfect powers m^k such that m^k = a^2 + b^4 = c^3 + d^5 for some positive integers a, b, c, d.

13140625, 36859543552, 49762009476, 87169610025, 3324163986441, 2988330556640625, 10155995666841600, 28920784535654400, 34328125000000000, 65388757868609536, 101445409544601600, 275625000000000000, 428123439576907776

Offset: 1

Author

Altug Alkan, Jun 07 2016

Keywords

• nonn

Comments

Intersection of A001597, A100293, A111925.

3625^2 = 13140625 is the least number with this property.

Sequence is infinite because if m^k = a^2 + b^4 = c^3 + d^5 is a term, then (m*t^60)^k = (a*t^(30*k))^2 + (b*t^(15*k))^4 = (c*t^(10*k))^3 + (d*t^(12*k))^5 is also a term for every t>1. - Giovanni Resta, Jun 07 2016

Examples

			13140625 is a term because 13140625 = 3625^2 = 2625^2 + 50^4 = 150^3 + 25^5.
		

Crossrefs

Cf. A001597, A100293, A111925.

Programs

PARI

isA111925(n)=for(b=1,sqrtnint(n-1,4), if(issquare(n-b^4), return(1))); 0
isA100293(n)=for(y=1, sqrtnint(n-1, 5), if(ispower(n-y^5, 3), return(1))); 0
list(lim)=my(v=List(), b4, t); for(e=2,logint(lim\=1,2), for(m=2,sqrtnint(lim,e), t=m^e; if(isA111925(t) && isA100293(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Jun 07 2016

Extensions

a(2)-a(13) from Giovanni Resta, Jun 07 2016