I was testing the properties of the Pythagorean triples of the form $(a,b=\frac{a^2-1}{2},c=\frac{a^2-1}{2}+1)$ and by chance I found that the following expression seems to be true for all the pairs (a,b):
If $((\varphi(b))^2$ mod $a$)$ + 1 = \tau(b)$ then $(a$ mod $\tau(b)) \in \Bbb P$
Where $\tau(b)$, also named $d(b)$, is the divisor count function, and $\varphi(b)$ is the totient function.
E.g.
$a=15 , b=112 , \varphi(112)=48 , \tau(112)=10$
$((\varphi(b))^2$ mod $a) + 1 = ((\varphi(112))^2$ mod $15) + 1 = 9+1 =10 = \tau(112)$ and $(a$ mod $\tau(b))=15$ mod $10 = 5 \in \Bbb P$.
This is the PARI/GP code just in case somebody wanted to try or modify it:
perct = 1;limit = 100000000;for(a=1,limit,if(a%2==0,if(a>((limit*perct)/1000),perct=perct+1;print("Current a",a));continue,b=((a*a)-1)/2;s1=length(divisors(b));totb=eulerphi(b);if(((totb*totb)%a)+1==s1,print(a," ",b," ",s1," ",totb," ",((totb*totb)%a)+1," ",a%s1))))
The rule seems to generate only primes, but (1) the set of primes get exhausted when $a \to \infty$ or (2) the distance between them grows up very quickly (exponentially?).
Tested so far in the interval $[1,56\cdot 10^6]$, the set of pairs $(a,a$ mod $\tau(b))$ is: $$\{(15,5),(29,5),(43,19),(449,71),(4909,13),(16171,43),(230741,149),(22267007,127)\}$$...no counterexamples, but it gets difficult to find new ones. OEIS does not show any sequence related to the $a$'s or the $(a$ mod $\tau(b))$ primes.
My theoretical knowledge is very basic, so I would like to share with you the following questions:
Although the relations in the expression seem too complex to discover the rule behind it, is there a reason for this property? is it trivial?
Can a counterexample be expected when $a$ gets bigger? I am still running the test but my computer is really slow, if somebody could give a try to the PARI/GP code, would be very appreciated.
Thank you!
Unfortunately after running the test the whole night I found a counterexample at $(118267641,57)$, so the property is not true.