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Learn the basics of number theory with free course materials from MIT OpenCourseWare: http://mitsha.re/dSFl30ct5lR

#MathMonday #Math #Maths #MIT #Education

#MathMonday #Math #Maths #MIT #Education

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I wanted to share my paper proposing an algorithmic proof on Goldbach's conjecture as well as the Twin Primes Conjecture. My proof is conceptually very similar to Eratosthenes sieve, but run in 2 parallel sieves. Look forward to your comments.

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**Prime Sieve**

The prime numbers appear in white squares. The multiples of prime numbers, then, are eliminated in ascending order until they are all removed - end up in coloured squares.

*Java programming credit: Lauren Williams*

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Øystein Ore:

In 1948 published:

"Number Theory and its History":

http://ia600806.us.archive.org/22/items/NumberTheoryItsHistory/Ore-NumberTheoryItsHistory.pdf

Thanks to:

https://ztfnews.wordpress.com/2013/08/13/oystein-ore-especialista-en-teoria-de-grafos/

In 1948 published:

"Number Theory and its History":

http://ia600806.us.archive.org/22/items/NumberTheoryItsHistory/Ore-NumberTheoryItsHistory.pdf

Thanks to:

https://ztfnews.wordpress.com/2013/08/13/oystein-ore-especialista-en-teoria-de-grafos/

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The following is a conjecture I formulated recently. It is strongly related to Goldbach's conjecture. Is this a genuine conjecture? Does it already exist? Can anyone find a counterexample? Your thoughts would be appreciated. I have an algorithm to test it (written in Python) but it requires optimisation. I will post the algorithm if anyone is interested.

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**prime generating polynomial 86 out 100: 38 in a row;47 + 5 x + x^2:**

I found this polynomial 23+5*x+x^2 this morning in an old experiment.

That polynomial has 58 primes in the first 100.

I thought to look and see if it was in a sequence like the Euler quadratic Heeger number polynomials

and found this new polynomial (at least to me).

It beats anything on this page:

http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

38 in a row

86 in the first 100

580 in the first 1000

(* Mathematica*)

In[439]:= Table[If[PrimeQ[47 + 5 x + x^2] == True, 47 + 5 x + x^2], {}], {x, 0, 100}]

Out[439]= {23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 289, 323, \

359, 397, 437, 479, 523, 569, 617, 667, 719, 773, 829, 887, 947, 1009, 1073, \

1139, 1207, 1277, 1349, 1423, 1499, 1577, {}, {}, 1823, 1909, {}, 2087, 2179, \

2273, 2369, {}, 2567, 2669, 2773, 2879, 2987, 3097, {}, 3323, 3439, 3557, \

3677, 3799, 3923, 4049, 4177, {}, 4439, 4573, 4709, 4847, 4987, 5129, 5273, \

5419, 5567, 5717, {}, 6023, 6179, 6337, 6497, {}, {}, 6989, {}, 7327, 7499, \

{}, 7849, {}, 8207, {}, 8573, 8759, 8947, 9137, {}, 9523, 9719, 9917, 10117, \

10319, {}}

In[442]:= Table[If[PrimeQ[47 + 5 x + x^2] == True, 47 + 5 x + x^2 {}], {x, 0, 38}]

Out[442]= {23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 289, 323, \

359, 397, 437, 479, 523, 569, 617, 667, 719, 773, 829, 887, 947, 1009, 1073, \

1139, 1207, 1277, 1349, 1423, 1499, 1577, {}}

In[443]:= Length[Delete[%, 39]]

Out[443]= 38

In[431]:= Delete[Union[

Table[If[PrimeQ[47 + 5 x + x^2] == True,47 + 5 x + x^2 {}], {x, 0, 100}]], 87]

Out[431]= {23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 289, 323, \

359, 397, 437, 479, 523, 569, 617, 667, 719, 773, 829, 887, 947, 1009, 1073, \

1139, 1207, 1277, 1349, 1423, 1499, 1577, 1823, 1909, 2087, 2179, 2273, 2369, \

2567, 2669, 2773, 2879, 2987, 3097, 3323, 3439, 3557, 3677, 3799, 3923, 4049, \

4177, 4439, 4573, 4709, 4847, 4987, 5129, 5273, 5419, 5567, 5717, 6023, 6179, \

6337, 6497, 6989, 7327, 7499, 7849, 8207, 8573, 8759, 8947, 9137, 9523, 9719, \

9917, 10117, 10319}

In[432]:= Length[Delete[

Union[Table[If[PrimeQ[47 + 5 x + x^2] == True, 47 + 5 x + x^2 {}], {x, 0, 100}]],

87]]

Out[432]= 86

In[433]:= Sum[If[PrimeQ[47 + 5 x + x^2] == True, 1, 0], {x, 0, 100}]

Out[433]= 86

In[444]:= Sum[If[PrimeQ[47 + 5 x + x^2] == True, 1, 0], {x, 0, 1000}]

Out[444]= 580

(* end*)

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