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due to the importance of the primes , the mathematicians have been investigating about them since long centuries ago . in 1801 , carl gauss , one of the greatest mathematician , submitted that the problem of distinguishing the primes among the non - primes has been one of the outstanding problems of arithmetic @xcite . proving the infinity of prime numbers by euclid is one of the first and most brilliant works of the human being in the numbers theory @xcite . greek people knew prime numbers and were aware of their role as building blocks of other numbers . more , the most natural question asked by human being was this what order prime numbers are following and how one could find prime numbers ? until this time , there have been more attempts for finding a formula producing the prime numbers and or a model for appearance of prime numbers among other numbers and although they could be more helpful for developing the numbers theory , however , the complicated structure of prime numbers could not be decoded . during last years , the prime numbers attained an exceptional situation in the field of coding . for example , `` rsa '' system is one of the most applicable system in this field used in industries relying on prime numbers . `` rsa '' system is used in most computerized systems and counted as main protocol for secure internet connections used by states and huge companies and universities in most computerized systems @xcite . on 2004 , manindra agrawal and his students in indian institute of technology kanpur could develop an algorithm called aks for detecting prime numbers @xcite . on 2006 , 2008 , 2009 and recently on 2013 , mathematics students in a project called detecting the mersenne prime numbers by computer network gimps succeeded to discover the greatest prime number . all such cases indicate the importance of mersenne theorem or any other approach for finding the largest prime numbers @xcite . generalizing the mersenne theorem , this paper could accelerate finding the largest prime numbers . in addition , there have been provided new equations and algorithm for attaining the largest primes . assume that @xmath0 is a natural number greater than 1 , @xmath1 related to n and natural numbers @xmath2 and @xmath3 are defined as below : @xmath4    if @xmath1 is a prime number , then @xmath0 is a prime number , too .    if @xmath0 is not the prime number so we can write @xmath0 as the multiplication of two natural numbers except @xmath5 meaning : @xmath6 @xmath7 @xmath8 @xmath9 @xmath10 @xmath11 therefore , @xmath1 is not the prime number . so , @xmath0 must be a prime number . this theorem is a generalization for mersenne theorem in which @xmath2 and @xmath3 are arbitrary natural numbers . if in the theorem @xmath12 , c is chosen as a multiple to @xmath2 and @xmath13 , thus , @xmath1 will not be a prime number . suppose : @xmath14 therefore : @xmath15 @xmath16 @xmath17 @xmath18 @xmath19 @xmath20 the last equality shows that @xmath1 is not a prime number . suppose @xmath0 is a natural number greater than @xmath5 , function @xmath1 related to @xmath0 and natural number @xmath2 are defined as below : @xmath21    if @xmath1 is a prime number , then @xmath0 is a prime number , too .    in this theorem @xmath22 , based on @xmath0 constant , please consider a sequence    @xmath23    we prove that sequence @xmath24 is strictly ascending , i.e. @xmath25 to prove the last inequality , we write : @xmath26 @xmath27 @xmath28 @xmath29 status 1 . if @xmath0 is a multiple of @xmath30 : @xmath31 status 2 . if @xmath0 is not a multiple of @xmath30 : @xmath32 therefore , inequity is accepted .    in this theorem , each number is higher than mersenne number , meaning : @xmath33 suppose @xmath2 be a natural number and @xmath34 are the primes smaller than or equal @xmath35 and @xmath36 , @xmath37 are natural numbers which limitations are intended for them indicated as follows : @xmath38    assume that @xmath39 is a function of @xmath40 which is displayed as bellow :    @xmath41    if the @xmath42 and @xmath37 circumstances are followed , @xmath39 can obtain all the primes less than @xmath2 . knowing that @xmath39 is odd , because it is non prime , therefore it comprises from two odd numbers except @xmath5 , and because @xmath43 , @xmath39 has at least a prime factor @xmath44 . therefore , @xmath39 is divided at least on one of the prime factors @xmath45 . @xmath46 @xmath47 it is clear that above equalities are in discrepancy of the assumption of the theorem .      1 . if : @xmath48 2 . interval @xmath49 : + it is clear that by putting minimum @xmath37 in the definition @xmath50 minimum @xmath51 followed by minimum @xmath39 is obtained as below : @xmath52    according to recent equation , it is obvious that being as prime number in prime numbers smaller than @xmath53 , r may not be divided into prime factors smaller than @xmath54 . on the other hand , it is not necessary to see if prime numbers smaller than @xmath53 are divided into @xmath55 to detect it as a prime number . indeed , for obtaining the prime numbers , we only require @xmath56 in @xmath57 to enter the provision of prime factor @xmath58 .      if @xmath59 is considered as a prime number bigger than @xmath60 , we could use @xmath61 instead of @xmath2 in this theorem because prime numbers smaller than @xmath35 include prime numbers smaller than @xmath62 . prime numbers smaller than 120 : @xmath63    \ { @xmath64 , prime numbers smaller than @xmath35 : @xmath65 } +    @xmath66    @xmath67    @xmath68    @xmath69    @xmath70    @xmath71    @xmath72    @xmath73    @xmath74    @xmath75 suppose @xmath2 be the natural number and @xmath76 are the primes smaller than or equal @xmath35 and also consider that @xmath77 are the primes larger than @xmath35 . suppose that @xmath78 and @xmath42 be the members of the natural numbers and also @xmath79 be the members of the account numbers , these variables are selected arbitrarily . function @xmath39 related to values @xmath80 , natural numbers @xmath81 and arithmetic numbers @xmath82 are defined as below : @xmath83 @xmath84    if @xmath56 be as the natural number less than @xmath2 , then @xmath39 is a prime number . if @xmath39 is not prime number , it has a prime factor @xmath85 . on the other side , because @xmath86 , @xmath39 has at least one prime factor @xmath87 . so , it is arbitrarily supposed that @xmath39 is divisible in @xmath88 . @xmath89 @xmath90    because @xmath54 is not denominator of any @xmath91 . we have : @xmath92 @xmath93 we reached a contradiction to the assumption . thus , the theorem was verified . we obtain prime numbers smaller than 119 . + @xmath94 @xmath95 @xmath96    table 1    [ cols="^,^,^,^,^,^",options="header " , ]     and continuing so suppose that : @xmath97 ( the order is considered in the primes )    general speaking , the theorem @xmath98 comes true to @xmath99 because @xmath100 includes the same primes . @xmath39 is obviously not divisible to @xmath101 and according to prime of the number @xmath39 , we have : @xmath102    to attain prime numbers , we divide the intervals as below : @xmath103 with regard to the relationship easier to be written . in example of the primes less than @xmath104 , the rang can be divided into three sections of @xmath105 , @xmath106 and @xmath107 . then , a distinct relation asserted for each . prime numbers smaller than 48 : @xmath108 @xmath109 @xmath110 @xmath111    and continuing so by integrating the relations , particularly using the relation 2 and notation 5.4 , we can attain an algorithm to obtain the largest prime number . ; one of them is as below :                          first of all , assign @xmath36 as one @xmath119 so as to obtain the minimum of @xmath0 . then in the following equation , give a counter to @xmath0 through the minimum @xmath0 so long as k be a member of the natural numbers @xmath120 . meanwhile , @xmath36 should not be even . @xmath121              if @xmath39 is not prime number , it has a prime factor @xmath85 . on the other side , because @xmath125 , @xmath39 has at least one prime factor @xmath126 therefore , with no interruption in the generality of the subject , we can assume that @xmath39 is divisible on @xmath127 : @xmath128 @xmath129 @xmath130 we reached a contradiction to the assumption . thus , the theorem was verified . 99 p. ribenboim , the little book of bigger primes , 2nd ed . springer science & business media , 2004 . w. stein , `` elementary number theory : primes , congruences , and secrets '' , book , pp . 1 - 172 , 2009 . r. crandall and c. b. pomerance , prime numbers : a computational perspective . springer science & business media , 2006 . m. agrawal , n. kayal , and n. saxena , `` primes is in p '' , ann . 781 - 793 , 2004 . `` list of known mersenne prime numbers - primenet '' . [ online ] . available : + http://www.mersenne.org/primes/ [ accessed : 08-apr-2015 ] .
today , prime numbers attained exceptional situation in the area of numbers theory and cryptography . as we know , the trend for accessing to the largest prime numbers due to using mersenne theorem , although resulted in vast development of related numbers , however it has reduced the speed of accessing to prime numbers from one to five years . 
 this paper could attain to theorems that are more extended than mersenne theorem with accelerating the speed of accessing to prime numbers . 
 since that time , the reason for frequently using mersenne theorem was that no one could find an efficient formula for accessing to the largest prime numbers . 
 this paper provided some relations for prime numbers that one could define several formulas for attaining prime numbers in any interval ; therefore , according to flexibility of these relations , it could be found a new branch in the field of accessing to great prime numbers followed by providing an algorithm at the end of this paper for finding the largest prime numbers .