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Is the conjecture on A+B=C following correct ?

Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write:

$A=a_1^{x_1}a_2^{x_2}...a_n^{x_n}$,

$B=b_1^{y_1}b_2^{y_2}...b_m^{y_m}$,

$C=c_1^{z_1}c_2^{z_2}...c_k^{z_k}$

Let $d = \min\{x_i, y_j, z_h \}$ where $1 \le i \le n, 1\ \le j \le m, 1\le h \le k$ then $$d \le 5$$

PS: I read above one hunded papers, I observed that in any case $\min\{x_i, y_j, z_h \} \le 3$

Example 1: Ten solutions of Catalan-Fermat equation

Example 2:

$2^4.3^5.7^6+5^9.11^8=19^1.23^1.47^1.6679^1.3051977^1$

See also:

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Đào Thanh Oai
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    If $\min{x_i,y_j,z_h}\geq 6$, then $rad(ABC)\leq \sqrt[6]{ABC}\leq C^{1/2}$, so $C\geq rad(ABC)^2$. The truth of the abc conjecture would imply there are only finitely-many triples $A,B,C$ with $\min{x_i,y_j,z_h}\geq 6$. – Julian Rosen Jun 19 '18 at 04:11
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    @JulianRosen: In fact Baker conjectured that $C<rad(ABC)^{7/4}$ which would mean that there are no solutions (for coprime $A,B,C$). – GH from MO Jun 19 '18 at 04:12
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    See ``A Local Version of Szpiro’s Conjecture'' by Bennett and Yazdani. – Pasten Jun 19 '18 at 04:28
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    @JulianRosen would $\min{x_i, y_j, z_h} \geq 5$ not imply that $C \geq \text{rad}(ABC)^{5/3}$, which would mean that $abc$ implies that there are only finitely many solutions? In fact wouldn't the even weaker assumption that $\min{x_i, y_j, z_h} \geq 4$ work as well? Even in the edge case when the minimum is 3, the exponents can't all be equal to three since that would violate Fermat's Last Theorem, so even in that case some more can be said – Stanley Yao Xiao Jun 19 '18 at 13:00
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    Of course, if you believe ABC (alas, it looks like there is a flaw in Mochizuki's proof, if I understand the latest news right), all such questions become simple exercises. But can we say something unconditional (or, at least, based on weaker conjectures)? – fedja Jun 19 '18 at 13:06
  • What do you think if $min{x_i,y_j,z_h} \le 3$ ? – Đào Thanh Oai Jun 20 '18 at 11:03
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    @fedja Where can I learn more about these "latest news" re Mochizuki's proof? Also, I understand from the above comments (I don't know anything about such things myself) that even ABC shows only the finiteness of numbers of solutions to this kind of equation: here, OP is conjecturing an absolute bound for a fixed exponent. – Gro-Tsen Jun 21 '18 at 10:50
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    @Gro-Tsen I think fedja refers to the following blog post http://www.math.columbia.edu/~woit/wordpress/?p=9871 and the links therein. – François Brunault Jun 21 '18 at 11:12
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    @fedja: "it looks like there is a flaw in Mochizuki's proof": Could you cite a news source? – Joseph O'Rourke Jun 21 '18 at 11:27
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    The number of trivial edits is getting ridiculous. Please don’t edit the question just to get it back on the front page. – Andy Putman Jun 26 '18 at 00:22
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    @AndyPutman I am sorry, I am not purposive – Đào Thanh Oai Jun 26 '18 at 00:46
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    There has been another completely trivial edit. Can you please stop fiddling with this question? If knowledgeable people are interested and have something to say, they will say it. It may be the case that people do not have anything useful to say, in which case you should accept this and move on – Yemon Choi Jul 08 '18 at 15:43
  • @YemonChoi I am sorry, thank You, because, I want correct edit paper to papers. I am sorry – Đào Thanh Oai Jul 08 '18 at 15:46
  • I am very happy if someone give to me an ideas to check this conjecture? – Đào Thanh Oai Jul 09 '18 at 03:52
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    The introduction now seems to ask “Can you comment on the following conjecture?”, which is off-topic; and also “Can you provide a reference for the following conjecture?”, which is unhelpful since we have no reason to believe that a reference settles it. So the post could use a better question, with a question mark at the end, like “Is the following conjecture correct?” –  Sep 16 '18 at 16:43
  • @MattF. Yes OK. I agree with You. Thank You very much, so please help me changes the question to "Is the conjecture on A+B=C following correct?". Because I don't want edited again myself(See comment byYemon Choi). – Đào Thanh Oai Sep 16 '18 at 23:24
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    @ĐàoThanhOai, given the strong votes in favor of no more changes, I will downvote this in its current form. –  Sep 17 '18 at 11:47
  • @MattF. I want changes but I need your help. Thank You very much – Đào Thanh Oai Sep 17 '18 at 14:12
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    Please stop bumping the question with edits – Yemon Choi Jan 03 '19 at 17:04
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    @YemonChoi I have modified it many times before, every time I fix it I save it immediately. I mean in some minute (10-20 minutes) I edit and save 10 times. During the time, the topic is not changes top list. This is my skill not good. Therefore I didnot edit to promotion. If I dited to promotion, some days I edited one time to top list. – Đào Thanh Oai Jan 03 '19 at 17:24
  • @YemonChoi When You give the first comment; You see the big number of edit, But maybe You didnot see I edited and saved edited and saved again in short time. So that time I edited, but I don't think I do that to promotion. I edit because my english, or my question is not clear, or not good. – Đào Thanh Oai Jan 03 '19 at 17:30
  • @AndyPutman Please see comment above – Đào Thanh Oai Jan 03 '19 at 17:36
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    @ĐàoThanhOai You have edited this question, consequently bumping it to the front page, on June 19, June 20, June 21, June 22, June 23, June 25, June 26, July 6, July 8, September 17, and now today; additionally, you have posted bounties on June 28, July 6, and September 12, each instance of which also bumping the question to the front page. Your claim that your edits are compressed into a short time period is simply false. Please refrain from further superficial edits. – Noah Schweber Jan 03 '19 at 18:05
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    The answer to the question appears to be unknown based on what users have said. I think many users here are asking you to please stop making edits. Making edits will probably not get your question answered. – Not a grad student Jan 03 '19 at 18:11
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    This whole constant editing thing is getting ridiculous. I've voted to close. – Andy Putman Jan 03 '19 at 18:27
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    @NoahSchweber I think You don't accepted my exculpation. But first I don't known how I can see the history of my edit until now. The edited in "June 19, June 20, June 21, June 22, June 23, June 25, June 26," is normal with one new memmber. I had joined 10 June. That time I don't known the Law. That time I don't agree with AndyPutman that "to get it back on the front page" – Đào Thanh Oai Jan 04 '19 at 03:00
  • The important that my english is not good – Đào Thanh Oai Jan 04 '19 at 03:01
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    I'm voting to close this question because Julien Rosen's comment seems to give the best possible answer and it is not clear what more the OP hopes to gain by all these continual edits – Yemon Choi Jan 05 '19 at 17:35
  • @YemonChoi https://mathoverflow.net/posts/3347/revisions – Đào Thanh Oai Jan 06 '19 at 23:17
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    DTO: in the question you link to, there are only 7 revisions in the space of 9 years, as opposed to 18 revisions you have made in 6 months. Moreover, in the revisions to my question, most of the edits were made by me within the first month of the original post, in direct and constructive response to requests for clarification, or to acknowledge actual new points raised by other people in comments or answers. Then 4 years later I addressed some points raised by Stefan Kohl, & since then I have not edited the question for another 5 years. This is not comparable to what you have been doing. – Yemon Choi Jan 07 '19 at 00:06
  • @YemonChoi I am sorry – Đào Thanh Oai Jan 07 '19 at 01:10
  • @YemonChoi I am sorry, but God be my witness; my edition on "June 19, June 20, June 21, June 22, June 23, June 25, June 26, July 6, July 8" I edit but I don't think to top list. I only edit to correct grammar. After that I bounty some times and last edit to top list – Đào Thanh Oai Jan 07 '19 at 01:19
  • I checked the conjecture 1 true with $1 \le a <b \le 10^8$ and $d \le 3$ – Đào Thanh Oai Sep 02 '19 at 15:49
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    For $d\ge 3$ the smallest example is $271^3+2^3 3^5 73^3=919^3$ (with underwhelming $abc$-quality of 1.10602). For $d\ge 4$ there are no solutions up to $10^{16}$. (Obviously this comment does not warrant editing the question.) – Yaakov Baruch Oct 05 '19 at 21:21
  • @YaakovBaruch I am sorry, why you kown that the conjecture true up to $10^{16}$ ? Do you compute? – Đào Thanh Oai Oct 06 '19 at 10:47
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    Yes, I generated all 4-powerful numbers up to that (easy, since each is a product of a 4th, 5th, 6th and 7th power), then looked for additive relations in that set. However I wasted time... Should have used instead a list of ALL abc-triples up to 10^18 to check which are powerful: http://www.math.leidenuniv.nl/~desmit/abc/abctriples_below_1018.gz . I'm currently searching there for $d\ge 3$ triples. There will be very few and of course unlikely any will have $d\ge 4$. I believe there is a complete list of triples all the way up to $10^{20}, but haven't found it. I use simple gawk -M commands. – Yaakov Baruch Oct 06 '19 at 12:45
  • @YaakovBaruch Thank You very much, Please see PS in topic. I hope that the property true with $d \le 3$ – Đào Thanh Oai Oct 06 '19 at 13:36
  • Thank You for your work @YaakovBaruch – Đào Thanh Oai Oct 06 '19 at 13:43
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    Indeed in the $abc$-list above (up to $10^{18}$) there just only one other powerful triple: $3^4 29^3 89^3+7^3 11^3 167^3=2^7 5^4 353^3$, with quality $q=1.08666$. – Yaakov Baruch Oct 06 '19 at 15:30

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