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March 7th, 2017, 05:10 AM   #1
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Post 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.

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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, 14-5 = 9, 27-14 = 13, 44-27 = 17,…….. If we form a list of these differences: (9,13,17….) we see that they form an AP with common difference 4.
• ]
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March 7th, 2017, 05:53 AM   #2
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$\displaystyle [a_1+(n-1)d_1][a_2+(n-1)d_2]=a_1a_2+(a_1d_2+a_2d_1)(n-1)+d_1d_2(n-1)^2$

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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