[warut]

เกร็ดเล็กๆน้อยๆเกี่ยวกับสิ่งที่ Hilbert เคยทำนาย (ผิดพลาด) เอาไว้ และนิสัยบางอย่างของ Hilbert จากหนังสือ "Hilbert" โดย Constance Reid หน้า 164 ครับ

"Siegel came to Göttingen as a student in 1919... he was always to remember a lecture on number theory which he heard from Hilbert at this time. Hilbert wanted to give his listeners examples of the characteristic problems of the theory of numbers which seem at first glance so very simple but turn out to be incredibly difficult to solve. He mentioned Riemann's hypothesis, Fermat's [Last] theorem, and the transcendence of $2^{\sqrt2}$ as examples of this type of problem. Then he went on to say that there had recently been much progress on Riemann's hypothesis and he was very hopeful that he would live to see it proved. Fermat's problem had been around for a very long time and apparently demanded entirely new methods for its solution - perhaps the youngest members of his audience would live to see it solved. But as for establishing the transcendence of $2^{\sqrt2}$ no one present in the lecture hall would live to see that!

The first two problems which Hilbert mentioned are still unsolved. [Fermat's Last Theorem ได้รับการพิสูจน์หลังจากที่หนังสือถูกตีพิมพ์ไปแล้วครับ] But less than ten years later a young Russian mathematician named Gelfond established the transcendence of $2^{\sqrt{-2}}$. Utilising this work, Siegel himself was shortly able to establish the desired transcendence of $2^{\sqrt2}$.

Siegel wrote to Hilbert about the proof. He reminded him of what he had said in his 1920 lecture and emphasised that the important work was that of Gelfond. Hilbert was frequently criticized for "acting as if everything had been done in Göttingen." Now he responded with enthusiastic delight to Siegel's letter, but he made no mention of the young Russian's contribution. He wanted only to publish Siegel's solution. Siegel refused, certain that Gelfond himself would eventually solve this problem too. Hilbert immediately lost all interest in the matter."