## Three Pearls of Number Theory by A. Y. Khinchin, Mathematics

By A. Y. Khinchin, Mathematics

Those three puzzles contain the facts of a easy legislation governing the realm of numbers identified to be right in all confirmed situations — the matter is to end up that the legislations is usually correct. Includes van der Waerden's theorem on mathematics progressions, the Landau-Schnirelmann speculation and Mann's theorem, and an answer to Waring's problem.

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2,\ till' "'"111 I n. «llall()JI" of Iype (27), I. , .. 1linn",. In thiS COIIII('('lion w(' "blainpr/ lillllt-; \\lthin \\hich thc ullkllu\\n::. JI"). \ certain ne\\ dIfficulty (the ra~ f()r thl' Iran"ltinn to lin'! 'JlHltlOlI") IInknn\\lls z. hU\e' Lo Le con-;jticre'd IIC\\ \\111. I'" r('r[aill I11UItI- rlil'itil's (fnr \\hie'h \\\: havc .. ( limits). ~ IllJlllbcrs Thl'rdorc \\l' IIHHle hi arc cho:scn ,lOti fixed. llpl}' Lhp rf>c;l\lt (lbtollllt,c1. b} Ihe 1111111- uer nf all 'illl h I'o". · dllll ce,...

In view of (28) u~n, lai,,,1 < c(n)N(u-l)fn. )1 n-u~ c(n) N( u-l )/n N(n-u)1 n=c(n)N(n-l )/n. The same estimate (with another c(n» holds for the whole ¢i(v;(j}), since the number of terms of this polynomial is equal to n. Accordingly 59 But every zi is the sum of 2 5 =c(n) summands of the form ±¢/vP'), and therefore ( .... ith another c(n). naturally). \'(n-I)/n>. Let m b(' nne of these numbers. 58) is such that one and the SdIne value of zi can very ..... ell result from different choices of the nllmhers vii> (1 ~j~2$).

Where c set ¢C and c' ¢ r:. Hut thpn ,Icrordi II).! tu the dcfini tion of the C"'. 1' .. II . ' sh,mn that 1 -t 8 1 ( ( : 1. Tu prove the imerse relation. -l thut pithPr c EC or c EC*. If c EC, tiwn c=u+b, a 1" eli l ' If. 1. bEN ho\~e\'er, c EC1, then, for a certain a E,I, the nllTliLer G-a. -l in U*. Jssumrtion, 7/ ¢c. It r: l' i:-. b*= ,-n 1 • l. , lIul appcar in the extension C 1. For if \\e had n EC*, \\P c01l1rl, by the definition of C"', put c'=n in equation (9), which would yipld C~1l1 {30 E I +-B=C, whereas c ¢C according to (9).