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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 013.10402

**Autor: ** Erdös, Paul

**Title: ** The representation of an integer as the sum of the square of a prime and of a square-free integer. (In English)

**Source: ** J. London Math. Soc. 10, 243-245 (1935).

**Review: ** The author proves the theorem that, if n is a sufficiently large integer, then primes p and quadratfrei integers f exist such that n = p^{2}+f when n\not\equiv 1 \pmod 4 and n = 4p^{2}+f when n\equiv 1 \pmod 4. The proof involves the prime-number theorem. The author states that he can prove similarly the theorem that n = p^{k}+q, where k is a given exponent and g has no k-th power as divisor.

Presumably for certain values of k there is an exceptional case corresponding to n\equiv 1 \pmod 4 when k = 2, but this is not stated; for example, if k = 4 and n\equiv 1 \pmod 16, n = p^{k}+g is not possible unless p = 2 and n-16 is k-th power free.

**Reviewer: ** Wright (Aberdeen)

**Classif.: ** * 11P32 Additive questions involving primes

11N25 Distribution of integers with specified multiplicative constraints

**Index Words: ** Algebra, number theory

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