Can anyone please verify/review this counterexample?

Thread starter Math100
Start date Feb 25, 2022
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Counterexample

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SUMMARY

The discussion centers on disproving the conjecture that every positive integer can be expressed in the form p + a², where p is either a prime number or 1, and a ≥ 0. A counterexample is provided using the integer 25, which can be expressed as 25 = 1 + 24, where 24 is neither prime nor 1. The participants emphasize the necessity of checking all possible expressions of the form p + a² to validate the disproof, highlighting that 24 is not a perfect square, thus reinforcing the argument against the conjecture.

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Feb 28, 2022

SammyS

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PeroK said:

Did you try ##34##?

Whether you you try n=34 or some other candidate as a counter example, try to be systematic.

To see if some integer, ##n## is a counterexample, begin with ##a=0## and increment ##a## by ##1## while ##a^2<n##. For each value of ##a##, compute ##p_a## where ##p_a=n=a^2##.
For a given number ##n##, if all ##p_a## are composite (Neither prime, nor 1), then you have found a counter example.

Furthermore, consider using as candidates for ##n##, non-prime numbers of the form ##n=3k+1##. . You should find that using such a number for ##n## will generate a list of values for ##p_a## in which 5 of every 6 entries is divisible by 2 or 3 (or both).