The Goldbach conjectures

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1 The Goldbach cojectures Jamel Ghaouchi To cite this versio: Jamel Ghaouchi. The Goldbach cojectures <hal > HAL Id: hal Submitted o 14 Dec 2015 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 The Goldbach cojectures Jamel Ghaouchi Abstract We deal with two problems kow by the ame of Goldbach cojectures, weak ad strog versios. The Goldbach cojectures They are two of the oldest ad best-kow problems of umber theory ad all mathematics. The first, kow as the weak oe, states that a odd umber greater tha 25 is always the sum of three prime umbers, whe the strog Goldbach cojecture states that a eve umber greater tha 22 is 28 always the sum of two prime umbers. This last oe has bee show to hold up through but remais uprove despite cosiderable effort. We preset here a very elemetary approach of the problems ad prove them by the same way. For the strog versio, we preset two approaches! The weak Goldbach cojecture First of all, a odd umber ca be differet of the sum of three powers of odds. We have 2k=4p+2m+1+2k'-1 that describes all eve umbers from 4p+2k' util ifiity whe m describes all the itegers. Thus 2k+1=4p+2m+1+2k' describes all odd umbers from 4p+2k +1 util ifiity for the same m. But 3p+2m describes all the odd umbers from 3p to ifiity whe m describes all the itegers. Hece, there always exists m for which 3p+2m'=q is a prime umber. We deduce that 2k+1=p+3p+2m'+2(m-m')+2k'+1=p+q+2(m-m')+2k'+1 Describes all the odd umbers from 4p+2k +1 util ifiity ad this for every k'. But 2(m-m')+2k'+1 describes all the odd umbers from 2(m-m )+1 to ifity whe k describes all the itegers. There always exists k =k for which 2(m-m )+2k +1=r a prime umber. I coclusio : 2k+1=p+q+r which describes the odds from 4p+2k +1 to ifity is always the sum of three prime umbers. We wat to calculate the first value of 2k+1 : Let p=5 the =2k+1 meas k=12=2p+k'=10+k' ad we kow ow that the cojecture is true for all the odds from 25 to ifiity! First approach The strog Goldbach cojecture

3 First of all, we must otice that a eve umber is ot always the sum of two powers of odd umbers. A couter example will show it : >1, m=1 (or m=2 or 3) ad the expoet =2 =2 6 p p p p p p 2 p p 0 3 p p p 1 Ad for p p p p 2 p p p p 2 p p 0 3 p p p 1 There is a eve umber (6) which is ot equal to the sum of the powers of two odd umbers! It meas that, i what follows, =1. For every eve umber 2m greater or equal to 12, there exists a iteger I, a prime p ad a iteger k for which : 2 m 4ip 2 k We wat to suppose there exists a eve which is ot equal to the sum of two primes. We must have q r 2 k 4 p N With q ad r are primes. Thus it is ratioal : q r 2 k a 4 p b WithGCD(a,b)=1 But 4 a p b ( q r 2 k ) b does ot divise a, ad as p is prime, b does ot divise p, hece it divises 4, cosequetly b=4 or b=2. If b=4 a p q r 2 k It meas aeve : impossible because GCD(a,b)=1 Thus b=2

4 q r 2 k 2 p a As i ca take every value : let i=a 2 m 4 ip 2 k 4 a p 2 k 2 q r 2 k Or m 2ip k 2 a p k q r k Ad q r 2 k 2 p a i ( 2 k ' 1) w k ', w N As 2 m 4 ip 2 k q r 2 k '' k w 2 ip k '' k It meas q r 2 2 w 2i Where w isiteger or q r 2 w 2 k Let w i 2 k ' 1 Ad q r 2 ( w 2 i) 2 2 ( 4 k ' 1) i 2 m ( 4 k ' 1) m ( 4 k ' 1) k 2 m m ( 4 k ' 1) k 2 k ' 1 p ( 2 k ' 1) p p ( 2 k ' 1) p Thus q r m 4 k ' 1 k 2 k ' 1 q r k 4 k ' 1 k q r 4 k ' k Hece doc 2 k q r

5 Or 2 m 2 ( q r ) 2 k q r But we have supposed the cotrary : it meas that it is impossible to suppose that there exists a eve is differet of the sum of two primes! Remark that q ad r ca be oly. We coclude that a eve is always the sum of two primes! Secod approach Also 2k=4p+2m+2k' describes all the eve umbers from 4p+2k' to ifiity whe m describes all the itegers ad, we saw it, there always exists m' ad q a prime umber for which 3p+2m'=q. Thus 2k=p+q+2(m-m')+2k' describes all the eves from 4p+2k to ifiity ad this for every k. Particularly, there always exists k =k for which : 2k'+2(m-m')=0 ad 2k=p+qdescribes all the eves from 4p+2k to ifity ad is the sum of two primes. Practically : For p=5 ad 5+17=2k or k=11=2p+k'=10+k'. It meas that the strog cojecture is true for all the eves from 22 to ifiity! Coclusio Our approach was sufficiet to demostrate the Goldbach cojectures. Bibliography [1] Fliegel, Hery F.; Robertso, Douglas S. (1989). "Goldbach's Comet: the umbers related to Goldbach's Cojecture". Joural of Recreatioal Mathematics 21 (1): 1 7. [2] Che, J. R. (1973). "O the represetatio of a larger eve iteger as the sum of a prime ad the product of at most two primes". Sci. Siica 16: