Suat Ayöz

Problem for June 16th from the 2026 AMS Daily Epsilon of Math Calendar

@DailyEpsilon

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Jun 16 at 6:00amWeb
Show threadbdesmonsJun 16
@PercyButtons3 @DailyEpsilon This problem made me think about an exercise I gave to my students this year: its statement (in French) is here:
https://manuel.sesamath.net/numerique/diapo.php?atome=88875&ordre=1
Let me mention @Sesamath which is doing a great job for us math teachers :)
Nombres premiers - Utiliser le théorème de Gauss appliqué aux nombres premiers - Exercice d’application n°48

Show threadAlex BoltonJun 16

@PercyButtons3 @DailyEpsilon Letting x = n², and (n + k)² = p + 17q, we can see that 2nk + k² = 10q, so k must be even.

Let k = 2m. Then (2n)(2m) + 4n² = 10q. So 2m(n + m) = 5q. But the left hand side is even, so q must be even, so q = 2.

So we are solving 2m(n + m) = 10. If m > 1 then LHS >= 2 × 2 × (n + 2) > 10 positive n. If m = 1 then 2(n + 1) = 10, so n = 4. This is the only possible solution. So x = 4² = 16 (and p = 2)