Unexpected output from Reduce?

Questions
,
1

In[86]:= Reduce[NextPrime - x == 2 && 0 < x < 20 && PrimeQ,
x [Element] Integers]

Out[86]= False

In[87]:= Reduce[NextPrime[x] - x == 2 && 0 < x < 20, 
 x \[Element] Integers]

Out[87]= x == 3 || x == 5 || x == 9 || x == 11 || x == 15 || x == 17

why Out[86]=False ?

I expect

Out[86]= x == 3 || x == 5 ||x == 11 ||x == 17
2

I suppose it is because PrimeQ is meant as a test for numbers. When given a symbol, such as in PrimeQ[x], it gives False.

I would use Element[x, Primes], which remains unevaluated when x is a symbol.

3

Well, some non-obvious things here (at least as far as I could find out):

  • PrimeQ[x] gets evaluated immediately to False, before any number can be even tried;
  • Reduce[] seems to expect a system of equations and inequalities only;

So - here is my approach:

primeQ[x_?NumericQ] := PrimeQ[x]
Reduce[NextPrime[x] - x == 2 && 0 < x < 20 && primeQ[x] == True, x \[Element] Integers]
4

Gianluca, your are simply too fast - and with the better solution! Best regards – Henrik