Are there solutions to $a^5=b^5+c^5+d^5$ where $a,b,c,d \in \mathbb{N}$?

Question

This is a generalization of Fermat's last theorem which states that there are no solutions to: $$a^n=b^n+c^n : a,b,c,n \in \mathbb{N}, n>2$$

A generalization of Fermat's last theorem, Euler powers conjecture: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture $$a_1^k+ a_2^k + \dots +a_n^k=b^k : n,k>1 \Rightarrow n \geq k$$

has been proven false, with the couter-examples for k=5 and 4. $$61917364224=27^5+84^5+110^5+133^5=144^5$$ and. $$31858749840007945920321=95800^4+217519^4+414560^4=422481^4$$

My question is in line of a more refined question related to Euler's conjecture, which is the smallest n for each k for which there is solution, or the equivalent formulation, the largest n for which there is no solution. In this line:

1. k=2 has solutions for n=2.
2. k=3 has solutions for n=3 and by fermat's theorem no solutions with n=2.
3. k=4 has solutions for n=3 and by fermat's theorem no solutions with n=2.
4. k=5 has solutions for n=4, what about n=3?
5. k=6 ...

Thank you.

Answer 1

According to 1967 paper by Lander et al (https://www.semanticscholar.org/paper/A-survey-of-equal-sums-of-like-powers-Lander-Parkin/f846f2247796ba411e724a1e4b1b02fbd74f12be) no solutions exist for numbers smaller than 2.6 * 10^14. I can't imagine (sadly) any realistic way of proving this statement true or false.