GENERALIZED ARITHMETIC PROGRESSIONS

Transcription

1 GENERALIZED ARITHMETIC ROGRESSIONS AND k Chun-Xuan an Bejn 00854,. O. Box 94,. R. Chna Lukx@publc.bta.net.cn Abstract We defne the eneralzed arthetc proressons and k, where,, k,,. If the arthetc functon 0, then there exst nfntely any pres such that,, are all pres. If 0, t has the fnte pre solutons, k AMS Matheatcs subject classfcaton: rary N.

2 . ntroducton We have proved that there are nfntely any and arbtrarly lon arthetc proressons of consecutve pres []. In ths paper we shall ove on to the study of the eneralzed arthetc proresson of consecutve pres. A eneralzed arthetc proresson of consecutve pres s defned to be the sequence of pres,,,,, k and,, k,, where s the frst ter, the coon dfference,. For exaple, 5,,,, and,, 4, s a eneralzed arthetc proresson of pres wth 5,, k and. Theore. The eneralzed arthetc proressons and k where j,,k,,. s called the We have the arthetc functon [-] ( χ( where, () χ s the nuber of solutons of conruence ), -th pre. () k ( q )( q ) 0(od ), () q,,. If χ ( ) and ( ) 0, then there are the fnte pre solutons. If 0, then there exst nfntely any pres such that,, are all pres. If ( ) 0, we have the asyptotc forula of the nuber of pres, k N [-] k N π k ( N,) ~, () k k k φ (lo N) where

3 φ ( ). To understand the eneralzed arthetc proressons theore, we yeld the ore detaled proofs below.. and k Theore.. and. Fro () and () we have () 0. It has no pre solutons f. Theore.. ( ) and k Fro () and () we have [-] ( ) 0. Snce ( ) as, there exst nfntely any pres such that, are all pres., Fro () we have [-] π ( N lo N N,) ~ 8φ If k 4, we have (5) 0. Theore.. 5( 0) and k 4 Fro () and () we have 0. There exst nfntely any pres such that,, 8 are all pres. If k 5, we have () 0. Theore.4. 4 ( 4 0) and k 9. Fro () and () we have 0. There exst nfntely any pres such that,, 8 are all pres. If k 0, we have () 0. Theore.5. 5 ( 5 0) and k 9. Fro () and () we have 0. There exst nfntely any pres such that,, 8 are all pres. If k 0, we have () 0. Theore.. and k. Fro () and () we have 0. If k, we have () 0. Theore.. and k 4. Fro () and () we have 0. If

4 k 5, we have (9) 0. Theore and k 5. Fro () and () we have 0. If k, we have () 0. Theore.9. 9 and k. Fro () and () we have 0. If k, we have (9) 0. Theore and k 4. Fro () and () we have 0. If k 5, we have () 0. Theore.. and k 9. Fro () and () we have ( ) 0. If k 0, we have () 0. Theore.. and k 4. Fro () and () we have ( ) 0. If k 5, we have (4) 0.. and k Theore.. and. It s called the eneralzed twn pre. Fro () and () we have where ( ) ( ) ( χ( )) 0, 5 χ f (od ); χ ( ) 0 f / χ ( ) otherwse. (od ); Snce as, there exst nfntely any pres such that and are all pres. Fro () we have N π ( N,) ~. φ lo N Theore.. and k, we have ( ) 0. If k 4 we have (5) 0. Theore.. 5 and k, we have 0. If k 4 we have () 0. 4

5 Theore.4. 4 and k 8, we have ( ) 0. If k 9 we have () 0. Theore.5. 5 and k 9, we have 0. If k 0 we have () 0. Theore.. and k, we have 0. If k we have () 0. Theore.. and k, we have 0. If k 8 we have (9) and k 9 Theore 4.. and. It s called the eneralzed twn pre. Fro () and () we have where ( ) ( ) ( χ( )) 0, 5 χ f (od ); χ ( ) 0 f / 9 (od ); χ ( ) 9 f f 9 (od ); χ ( ) 0 f / (od ); χ ( ) otherwse. Snce as, there exst nfntely any pres such that and are all pres. Theore 4.. and k, we have ( ) 0. If k 4 we have (5) 0. Theore and k, we have 0. If k 4 we have () 0. Theore and k 8, we have ( ) 0. If k 9 we have () 0. Theore and k 0, we have 0. If k we have () 0. Theore 4.. and k 5, we have 0. If k we have () 0. 5

6 REFERENCES. C. X. an. Foundatons of Santll s sonuber theory wth applcatons to new cryptoras, Ferat s theore and Goldbach s conjecture. Internatonal Acadec ress, 00. WWW. -b-r.or.. C. X. an. On the Yu-Goldbach pre theore. Guanx Scences (n Chnese), (99) 9-. CM C. X. an. Foundatons of Santll s sonuber theory. Alebras, Groups and Geoetres 5 (998) 5-9. MR000c: C. X. an. Foundatons of Santll s sonuber theory. Alebras, Groups and Geoetres 5 (998) CM C. X. an. Foundatons of Santll s sonuber theory. In: Foundaental open probles n scences at the end of the llennu, T. Gll, K. Lu and E. Trell (Eds) Hadronc ress, USA, (999) MR 00c: 09; MR 000k: 0004; ZM C. X. an. roof of Schnzel s hypothess, Alebras Groups and Geoetres 8 (00) CM C. X. an. re theore n Santll s sonuber theory a b. Alebras Groups and Geoetres 9(00) n n ± CM C. X. an. re theore s Santll s sonuber theory (II). To appear. 9. C. X. an, Dophantne equaton any pre soluton. To appear. λ n n n n λ λn b b has nfntely 0. C. X. an. Santll s pre chans: j aj ± b. To appear.. C. X. an. roof of the Batean-Horn conjecture. To appear.. C. X. an. Ffteen consecutve nteers wth exactly k pre factors. To appear.. C. X. an. Generalzed quadratcs representn pres. To appear.