THE AL AFLITUN’S 148th PROBLEM IN NUMBER THEORY
Sequences of products of 2 different odd primes (2-factor-n-ads)
We consider (see also the 59th Problem in ARITHMETIC, vol.1) a sequence of products of two different
odd prime factors
with examples:
1) n=5: 2-factor-5-ad (the pentad): 3*71=213, 5*43=215, 7*31=217, 3*73=219, 13*17=221;
; q_1=3,;; q;_2 =5,; q;_3=7,q_4=3,q;_5=13,; p;_1=71,p;_2=43,; p;_3 =31,; p;_4=73,; p;_5=17.
2) n=8: the 2-factor-8-ad (the ogdoad found by M. Polegendre 31/05/2021): 11*739=8129, 47*173=8131,
3*2711=8133, 5*1627=8135; 79*103=8137,3*2713=8139, 7*1163=8141,17*479= =8143 ; ;q_1=11,;; q;_2 =47,; q;_3=3,q_4=5,q;_5=79,; ; q;_6 =3,; q;_7=7,q_8=17,p;_1=739,p;_2=173,
p_3=2711,; p;_4=1527,; p;_5=103;,p;_6=2713,; p;_7=1163,p_8=479.
3) n=8: the 2-factor-8-ad (the ogdoad found by M. Polegendre 17/01/2023): 9983=67*149, 9985=5*1997,
9987=3*3329, 9989=7*1427, 9991=97*103, 9993=3*3331, 9995=5*1999, 9997=13*769;
;; q;_1=67,;; q;_2 =5,; q;_3=3,q_4=7,q;_5=97,; ; q;_6 =3,; q;_7=5,q_8=13,p;_1=149,p;_2=1997,
p_3=3329,; p;_4=1427,; p;_5=103;,p;_6=3331,; p;_7=1999,p_8=769 .
Since the search for subsequent similar structures becomes extremely difficult we need to computerize this
search process. The following algorithm is proposed for search starting from n=9 (from an ennead) (see
also The 144th Problem).
1) From the table of prime numbers we select sequences of pairs of consecutive prime numbers with
intervals equal to 20 and more between them.
2) In each of these intervals we decompose all composite numbers into prime factors.
3) From obtained results we leave only those in which sequences of more than 8 products of 2 different
prime numbers are found with a step equal to 2 between these products (the “sieve” principle).
4) Screened remaining sequences of more than 8 products of 2 different prime numbers are sent for printing.
THE 148th PROBLEM
Find a maximal n for 2-factor-n-ad.
How many 2-factor-9-ads (enneads) are there?