Congettura di Andrica

La congettura di Andrica è una congettura della teoria dei numeri, riguardante gli intervalli tra due successivi numeri primi, formulata dal matematico romeno Dorin Andrica nel 1986. Afferma che, per ogni coppia di numeri primi consecutivi pn e pn+1, si ha

Se poniamo , allora la congettura può essere riscritta come

semplicemente spostando a destra ed elevando entrambe le quantità al quadrato.

La congettura è stata verificata empiricamente per tutti i numeri primi minori di 1016.[1]

Una generalizzazione della congettura è lo studio dell'equazione

Si pensa che il più piccolo x per cui questa equazione sia risolubile sia [2] (la costante di Smarandache); per questo numero, i primi coinvolti sono p30=113 e p31=127.

Conseguentemente, la congettura generalizzata di Andrica afferma che per ogni x minore di questa costante la disequazione

vale per ogni n.

Note

  1. ^ Wells David, Prime Numbers: The Most Mysterious Figures in Math, Wiley, 18 maggio 2005, ISBN 978-0-471-46234-7.
  2. ^ La parte decimale di questo numero è la sequenza A038458 dell'OEIS

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