In[]:=
maxSize=150;(*listoffifthpowersofintegersfrom-maxSizetomaxSize*)basis=Range[-maxSize,maxSize]^5;(*thesearebynomeansuniquesolutions.IjusttakethefirstsolutionthatIntegerPartitionsfinds*)soln2025=IntegerPartitions[2025,{5},basis,1][[1]];soln2026=IntegerPartitions[2026,{5},basis,1][[1]];(*thelistofintegersthatgetraisedtothepowerof5andsumto2025and2026respectively*)soln2025=Sign[soln2025]*Abs[soln2025]^(1/5)soln2026=Sign[soln2026]*Abs[soln2026]^(1/5)
Out[]=
{100,51,44,11,-101}
Out[]=
{121,28,4,-79,-118}