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 = p2+f when n\not\equiv 1 \pmod 4 and n = 4p2+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 = pk+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 = pk+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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