Positive integers greater than 79 that are the sum of fourth powers of 19 positive integers.(minimum 19 terms)

Question

I know that numbers till 79 can be expressed as a sum of fourth powers of 18 positive integers and 79 is the smallest number to require 19 terms.

What are some other numbers that require a minimum of 19 terms? Repetition is allowed

Answer 1

Here's a snippet from "Waring's Problem" by P.M. Batchelder, in the Monthly, Vol. $43,$ No. $1$ (January $1936$) listing $6$ other numbers that require $19$ fourth powers. Of course, nowadays it is known that $g(4)=19.$

I wonder if the largest integer requiring $19$ fourth powers is known. YES

Answer 2

Well:

$$\sum_{n=1}^{18}n^4=432345$$ that is $1^4+2^4+\cdots+18^4=432345$, which disproves your theory nicely.

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For things requiring $19$ terms, we would have $$\sum_{n=1}^{19}n^4=562666$$

Fermat's Last Theorem (or some number-crunching) verifies that this can't be condensed into $18$ terms.

Furthermore, because Fermat's Last Theorem proves that a sum of $n$ fourth powers cannot be expressed as the sum of $n-1$ fourth powers (provided one of the powers isn't $0$), you can generate any list of $19$ non-zero fourth powers and their sum is certainly not expressible as the sum of $18$ fourth powers.