ON NUMBERS REPRESENTABLE AS THE SUM OF FOUR SQUARES OF PRIME NUMBERS

In this work, an new estimate is obtained for the number of representations of a sufficiently large natural number as a sum of squares of four prime numbers. Additionally, the cardinality of the set of natural numbers that cannot be represented in this form (the exceptional set of the problem) is estimated. The obtained result, in a certain sense, improves and complements previously known results in this field. The proof of the main result utilizes the Hardy–Littlewood circle method in the variant of I.M. Vinogradov.