en.wikipedia.org/wiki/Euclid_number
1 correction found
Likewise, the last digit of every Kummer number is 9.
This overstates the pattern: the first Kummer numbers are 1 and 5, so not every Kummer number ends in 9. The last-digit pattern 9 only begins once the primorial includes both 2 and 5.
Full reasoning
A Kummer number is defined here as (p_n# - 1), where (p_n#) is the product of the first (n) primes.
That means:
- for (n=1), the Kummer number is (2-1=1), which ends in 1;
- for (n=2), the Kummer number is (2\cdot3-1=5), which ends in 5.
So the claim that every Kummer number ends in 9 is false. The pattern becomes true only from (n \ge 3), because from that point the primorial contains both 2 and 5 and is therefore divisible by 10, making (p_n#-1) end in 9.
OEIS sequence A057588, which lists Kummer numbers, begins 1, 5, 29, 209, 2309, ..., directly showing the counterexamples 1 and 5.
2 sources
- OEIS A057588 - Kummer numbers
Kummer numbers: -1 + product of first n consecutive primes. 1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, ...
- Primorial - Wikipedia
The primorial p_n# is defined as the product of the first n primes ... The first few primorials p_n# are: 1, 2, 6, 30, 210, 2310, 30030, ...