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March 7th, 2017, 05:10 AM  #1 
Newbie Joined: Mar 2017 From: Earth Posts: 1 Thanks: 0  Segregation of Arithmetic Progressions
Came upon this interesting pattern while studying arithmetic progressions regarding their grouping like natural numbers are grouped into prime or composite, only this time there are 3 groups. If you could please have a look, see if it's worth it, and possibly tell me where to go next.  The APs we are considering in the following research are limited to a specific type, particularly : • APs consisting of natural numbers APs can be divided into three types: • Prime APs • Constant APs • Composite APs They are defined as follows: • Prime APs: o Cannot be divided into two APs other than (1,1,1….) and the AP itself. o Can consist of both prime and composite numbers. o Cannot consist of only composite numbers. o Common difference should be greater than 0. o Multiplication of corresponding terms of 2 prime APs will not result in a new AP. o Examples of prime APs: • (1,2,3,4…..) • (5,7,9,11….) • (3,7,11,15….) • (99,101,103,105….) • (2,5,8,11….) • Constant APs: o Common difference is always equal to 0. o Can only consist of prime numbers. o Multiplication of corresponding terms of 2 constant APs will always result in a new AP. o Examples of constant APs: • (3,3,3….) • (101,101,101….) • (53,53,53….) • (89,89,89….) • Composite APs: o Can be divided into 2 or more APs other (1,1,1…) and itself. o Can only consist of composite numbers. o When factorized or divided into two or more APs other than (1,1,1…) and itself, 1 or none of the resultant APs will be a prime AP, the rest will be constant APs. o Common difference can be greater than or equal to 0. o Multiplication of corresponding terms of a prime and constant AP will result in a composite AP. Only 1 prime AP can be used in the multiplication while the number of constant APs can be infinite. o Multiplication of corresponding terms of 2 or more constant APs will result in a composite AP. o Examples: • (2,4,6,8….) = (1,2,3,4…)x(2,2,2,2…) • (88,88,88…) = (2,2,2….)x(2,2,2…)x(2,2,2…)x(11,11,11…) • (9,15,21,27…) = (3,5,7,9…)x(3,3,3,3…) • (99,99,99…) = (3,3,3…)x(3,3,3…)x(11,11,11…) Therefore we can observe how two different types of APs can result in a third type of AP. All APs consisting of natural numbers can be classified into these three groups. • [EXTRA: (property) o If we take two prime APs and multiply their corresponding terms, the resulting list of numbers will not be an AP, but if we write down the difference between every 2 consecutive terms, that list will form an AP. o Example: • (1,2,3,4…)x(5,7,9,11…) = (5,14,27,44….). now taking the difference between ever two consecutive terms, 145 = 9, 2714 = 13, 4427 = 17,…….. If we form a list of these differences: (9,13,17….) we see that they form an AP with common difference 4. • ] 
March 7th, 2017, 05:53 AM  #2 
Newbie Joined: Jun 2016 From: Hong Kong Posts: 19 Thanks: 1 
$\displaystyle [a_1+(n1)d_1][a_2+(n1)d_2]=a_1a_2+(a_1d_2+a_2d_1)(n1)+d_1d_2(n1)^2$ the product of two polynomial with degree 1 is a polynomial with degree 2 $\displaystyle f(x)=ax^2+bx+c$ $\displaystyle \Delta f(x)=f(x+1)f(x)=a(2x+1)+b=2ax+a+b$ the Finite difference of the polynomial with degree 2 is a polynomial with degree 1 That's why the property exists 

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arithmetic, arithmetic progressions, progressions, segregation 
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