(a ^ 2 + b ^ 2) / (ab + 1) butun sonning kvadratini isbotlash uchun, biz uning butun son va kvadrat ildizi butun son ekanligini ko'rsatishimiz kerak. Avvalo, (a ^ 2 + b ^ 2) / (ab + 1) butun son bo'lishi mumkin, agar va faqat agar (a ^ 2 + b ^ 2) - k ^ ab + 1) = 0 ba'zi butun son k uchun. Qayta tuzatish, biz: a ^ 2 - kab + b ^ 2 - k = 0 a-da kvadrat tenglamasi sifatida ko'rib, diskrimintant mukammal kvadrat bo'lishi kerak, ya'ni: k ^ 2 - 4 ^ ((b ^ 2 - k) = m ^ 2 ba'zi butun sonlar uchun m. Soddalashtirish, biz: k ^ 2 - 4 ^ 2 - 4 ^ 2 - 4 ^ 2 - u = u = u = u =
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