Template:Sequence of the Day for November 18

Intended for: November 18, 2013

Timetable

• First draft entered by Alonso del Arte on October 17, 2012 ✓
• Draft to be reviewed by September 18, 2013
• Draft to be approved by October 18, 2013

Yesterday's SOTD * Tomorrow's SOTD

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A060003: Odd numbers not of the form

p + 2 b 2

for

p

prime and

b > 0

.

{ 1, 3, 17, 137, 227, 977, 1187, 1493, 5777, 5993, ¿...? }

On November 18, 1752, Christian Goldbach wrote a letter to Leonhard Euler in which he conjectured that every odd integer can be expressed as the sum of a prime and twice a square. Euler verified this lesser known conjecture of Goldbach’s up to 2500 and found no counterexamples. Goldbach did allow the square to be zero and considered 1 a prime number, and thus 3, 5, 7 are taken care of with 0 2, 9 = 1 + 2  ×  2 2 or 7 + 2  ×  1 2, etc.

When Moritz Stern read the Goldbach–Euler correspondence, he became interested in this problem and checked up to 9000, finding the composite numbers 5777 and 5993. With 5777, we can now quickly verify that not only is each

5777  −  2 b 2

composite, quite a few of them are not squarefree. And if we considered 1 prime like they did back then, it would not help here, since 5776 = 2 4  ×  19 2. By requiring

b > 0

, a prime needs a smaller prime for its representation as

p + 2 b 2

. For the primes in this sequence, now known as the Stern primes, there is no smaller prime such that the difference is twice a square.

With modern computers, M. F. Hasler and Benjamin Chaffin have verified there are no more terms up to 2  ×  10 13.