By Euler L.

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Chapter 4 ROTH'S THEOREM ON SQUARE - FREE INTEGERS Our remark (1') of the previous chapter shows that denoting by qn the nth square free number (in the order of magnitude) one has ^ \ An analytical proof of an improvement of this relation with 1/3 in place of 1/2 was later obtained by K. F. Roth. It was pointed out by T. Estermann (see. K. F. Roth; J. Lond. Math. Soc, (2) 26 (1951) pp. 263-268) that the method was equivalent to a simple and elementary argument. Now we formulate this result as a theorem and give the latter proof.

E. CRANDALL, E. W. MAYER and J. S. PAPADOPOULOS. The twenty-fourth Fermat number is composite, Math. ] Here the authors prove that F24 is composite thus completing that Fn = 2 2 " + 1 is composite for 5 < n < 32. A perfect number m (a positive integer) is one for which the sum of the divisiors including 1 and m is 2m. Some even perfect numbers are known. But it is conjectured that odd perfect numbers do not exist. In the same volume viz volume 72 (July 2003) pages 1549-1554, P. M. JENKINS has shown that an odd perfect must be divisible 26 THEORY OF NUMBERS by a prime exceeding 107 (senior thesis, Brigham Young University (2000)).

F. §4 of chapter 9. * see chapter 8, # t see chapter 9 (6)... SIMPLE Q RESULTS BASED ON SIMPLE PROPERTIES <(s) 33 proof. , for any e ^ 0 Y^ M»)l = ex 1/2 n —x is true for all x ^ XQ(= Xo(e) ^ 1). This implies that (5) holds in a > \ and so £(s) has no zero in a > | . In view of the result mentioned earlier this means that there is a zero of £(s) having real part = | . Let so = \ + #o be this zero. If m is the order of the zero so we have > ICWI 1 r^r = 1 T= 777; as s = a H- z^o -> 5 0 + 0. l*-so|m ~ l«-*o| *-l/2 with > - constant absolute.

### On amicable numbers by Euler L.

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