Here is a proof that Terry Tao's Green-Tao theorem is a fakery, but do what you have always done Terry-- run, hide, run, and hide.

Proof that the Green-Tao so called theorem is a con-art fakery of math, for the authors of Green and Tao never Well Define the concept of Infinity and that is the source of their failed proof
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Archimedes Plutonium
Aug 16, 2021, 12:58:22 AM (yesterday)
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Archimedes Plutonium
Aug 15, 2021, 7:56:54 PM
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AP writes: ..Notices how failed is Dr. Tao of UCLA with his con-art of the Green-Tao fake theorem, never defining what the hell is "infinity" and just plain assuming a infinity means "endless" when in fact as AP shown from Huygens that there is a Well Defined Infinity, provided one has a logical brain.

So, well there are striking similarities with Hales con-art fakery of Kepler Packing and the con-art fakery of Green and Tao, for both groups fail to well define infinity. And the impact on Hales is there is a more dense packing when you approach infinity borderline than the pure hexagonal close pack, a hybrid packing is the most dense with the bulk being HCP. So we see how a well defined infinity impacts the fake proof of Hales.

But how does that well defined infinity impact the Green - Tao con-art??

Well, it impacts them more than Hales's con-art.

Because the Huygens Tractric well defined infinity borderline is 1*10^604 and so, somewhere along the line of strings of primes separated by a specific distance length such as 4 for 3, 7, 11 for a string of 3 primes and 12 distance for 5, 17, 29, 41, 53 for a string of 5 primes. So what Tao and Green think they proved is that you can have a infinite number of primes for a string of a distance separation. But can you really if you well define infinity, what the hell is infinity? That you have to have WELL DEFINED CONCEPTS before doing proofs involved with infinity.

So, if the borderline of Infinity is 1*10^604. At what distance length will you cease having a string of consecutive primes before reaching 1*10^604? This is a question totally foreign to Green and Tao with their impish understanding of Infinity.

Now here, AP is going to conjecture based on a string of 3 takes up to 11, and a string of 5 takes up to 53. That AP is going to conjecture that a string of 1*10^604 primes separated by a distance of 1*10^604 is going to take up to about 10^1211. Far beyond the permissible limit of the 1*10^1208 allowed. And far exceeding the number 1*10^607-- going past the infinity borderline to get 1^10^604 primes between 1 and 1*10^1208.

I doubt any of our modern day computers can verify any of this. For I doubt any modern day computers can ever list list all the primes between 1 and 10^604.

But then in AP's teachings, why even bother? Why bother at all, because Primes is a ILL conceived set for there is no division operation on a set of counting numbers divided by other counting numbers-- for a well defined set always yields more members of that original set. Example 1/2 is not a counting number. But addition and multiplication are well defined over Counting Numbers. So here we have the question of why ever even bother with silly stupid ruminations about primes numbers? Why? For primes are exercises for the mentally ill in mathematics.

And this goes to show, that one con-art teaches another con-art-- Hales Kepler Packing teaches Green-Tao con art.

AP
King of Science, especially Physics
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Archimedes Plutonium
Aug 15, 2021, 8:58:25 PM
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And, because they are so far apart, beyond 1*10^1208, means the Green-Tao theorem is totally false, for there are no infinite strings of Primes separated by a finite distance. If a finite distance of 1*10^604 yielded a infinite string of primes of 1*10^604 between 1 and 1*10^1208, then the Green-Tao theorem is correct. But that is not what happens.

Take the case of pretend 10 is the infinity borderline, then 100 would be the ultimate limit for saying there are infinite numbers of so and so. Can we find a string of 10 primes separated by a distance of 10 given only 1 to 100? No, I see in the first 100 primes no string of 10 primes separated by any distance.

So, here, in these two posts, I have proven the Green-Tao alleged theorem is b.s. Total b.s. because the concept of "infinity was ill defined". And on top of ill defined infinity, grown ups playing with a ill defined set-- the Primes since Counting Numbers are ill-defined over Division.

So what does that make Tao and Green? Certainly not mathematicians.

Archimedes Plutonium
Aug 15, 2021, 9:55:07 PM
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No, it is not worth writing a whole new book for AP spelling out the b.s. failure of the Green-Tao theorem. No, instead this should be included in the Hales's Kepler Packing failure for both groups screwed up horribly with no Well Defined Infinity. And to think one can prove something about infinity when they are nerdy fools of even knowing what infinity means. There is an Aesop fable type of message and theme in all of this. That vagueness of a subject can never lead to clarity, but only more vagueness.

So this has my interest perked, to see what the maximum string of primes is from 1 to 100 if 10 were the infinity borderline. What I need is for the first string of 10 primes spaced a specific distance apart. And this tells everyone in mathematics, going forward, that all questions, yes all questions of is it infinite or is it finite has to go through this same process, "Are there 1^10^604 such things between 1 and 1*10^1208, the line of completeness." Primes, even though ill-defined themselves are infinite because there are 1*10^604 primes between 1 and 1*10^607 (this numbers needs sharpening up).

--- quoting utm.edu ---
The First 1,000 Primes
(the 1,000th is 7919)
For more information on primes see https://primes.utm.edu/

2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67 71
73 79 83 89 97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349
353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463
467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601
607 613 617 619 631 641 643 647 653 659
661 673 677 683 691 701 709 719 727 733
739 743 751 757 761 769 773 787 797 809
811 821 823 827 829 839 853 857 859 863
877 881 883 887 907 911 919 929 937 941
947 953 967 971 977 983 991 997 1009 1013
1019 1021 1031 1033 1039 1049 1051 1061 1063 1069

The maximum string I see is a distance separation of 6 and involves 5 primes, 5, 11, 17, 23, 29.

And the AP well defined infinity allows us to always use Mathematical Induction for proofs on whether finite or infinite.

And looking in the above, I cannot even spot a string larger than 5 in the first primes in 1 to 1000. So where is the first location of a string of 10 primes separated by a given constant distance? It must be a distance number far larger than 10 itself.

No, this news does not deserve a book titled Green -Tao have a screwed up theorem which is in fact false. We do not write science books on failed science. So it must be included in Kepler Packing Problem proof of AP.

And well, the Green-Tao failure is not over with, for there is the question of what duped both Green and Tao. What did they employ for their fake proof? And I would certainly bet that they used a Reductio Ad Absurdum, perhaps two or three RAA in their con art fakery.

AP, King of Science, especially Physics

Archimedes Plutonium
Aug 16, 2021, 12:42 AM
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--- quoting from the Web a site by Njwildberger who found a string of 10 primes separated in length by the constant 210 and they are 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089.

--- end quote ---

So that saves me a lot of time looking.

And it goes to show a fine example of a set that is not infinite, but is a finite set. So counter to the Green-Tao theorem, theirs is a fakery and theirs is false.

You cannot have a arbitrary long sequence of primes given any constant finite number.

The very definition of Infinity is it has a borderline at 1*10^604 due to the proof of Huygens about tractrix where the area of tractrix catches up with associated circle area at this number-- the number where pi has 3 zero digits in a row.

Once the borderline is established, any number larger is a infinite-number, any smaller or equal to is a finite number.

So to prove any set is an infinite set goes through the very same program of algebraic completeness at 1*10^1208. Are there 1*10^604 primes existing between 1 and 1*10^1208, yes and quite easily. So primes are an infinite set..

How about perfect squares? The set 1, 4, 9, 16, . . .? Glad you asked for that is a Minimal Infinite set for there are precisely 1*10^604 perfect squares between 1 and 1*10^1208.

What about the Green-Tao offering. The set of sequences such as sequence of 2 primes, sequence of 3 primes, sequence of 4 primes, .. sequence of 10 primes, etc, etc. Are there 1*10^604 of these sequences with every number from 1 to 1*10^604 covered by a sequence between 1 and 1*10^1208 ??

And the answer is clearly no, because the program of math induction of all the scale numbers, 1, 10, 100, 1000, etc clearly shows that there is not even sufficient numbers if we pretend 10 is the infinity number, causing us to show that the numbers from 1 to 100 has a sequence of 10 primes. The largest sequence I can find in 1 to 100 is a sequence of 5 primes where 5, 11, 17, 23, 29 separated by a constant of 6.