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I am searching for a mathematical conjecture that is true until something like n = 117, or n = 127, or a number close.

It is an equality, a formula implying calcultations to the power of n, IIRC.

What is the name of this conjecture ?

Why do I search for this formula ?:

I had read an article a few years ago that was discussing about experiments around the validation of a theory in Science, namely in Physics.

The example of this conjecture was given so as to convince that you can choose to repeat an experiment with different parameters a certain number of time, ultimately in the end, it does not give the entire certainty a physical theory is true: it just happens we still haven't found any outlier measurement that contracdicts the physical theory.

As a matter of fact, we can test this mathematical conjecture for the first 127 integers, which is a bunch of times, and believe it is always true, whereas it is false for all n.

I would like the name of it so as to be able to name it / source it in discussion.

Does it ring any bell for you ?

Stephane Rolland
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    Look here https://math.stackexchange.com/questions/111440/examples-of-patterns-that-eventually-fail and here https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Guy697-712.pdf – twnly Jul 11 '19 at 09:45
  • Like the conjecture that can easily be proved by induction that $;n\le116;$ ? Or also that $;n^2-117n<0;$ ?... – DonAntonio Jul 11 '19 at 09:45
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    Also see https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples/1145#1145 – Ethan Bolker Jul 11 '19 at 09:50
  • You can modify the Borwein integral to get something to this effect. – Jürgen Sukumaran Jul 11 '19 at 10:01
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    Although I have not recognized at first look the conjecture I had in mind, all of the resources you pointed me to are answering my need. Now I need to choose the most explicit example. I tend to think I will choose this one: https://math.stackexchange.com/a/365881/6235 . n17+9 and (n+1)17+9 are relatively prime The first counterexample is n=8424432925592889329288197322308900672459420460792433 – Stephane Rolland Jul 11 '19 at 10:28

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Well, we would have to define what exactly counts as a "conjecture". You can find a trivial example in something like:

"I conjecture that every positive integer can be expressed uniquely by 7 binary digits", but I guess this is not valid, so more rules should be specified.

If we need it to be about powers, then "I conjecture that every positive integer can be expressed uniquely by 127 binary digits"

David
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  • Not at all the conjecture I was searching for: But it is sort of a really good example nonetheless!!! Imagine an algorithm using binary representation of integers on one byte as you said, and the law/function/operation f(x)=x+1. Without testing more than 127 values, one cannot guess the the Law is cyclical, or that it has a singularity like 1/0 higher, or could even be topped/constant for any number > 127. In any case, the validity of the intuition that the Law always gives the upper integer is wrong, the intuition only cannot justify to make it a rule for all n. Great simple example – Stephane Rolland Jul 11 '19 at 11:06