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Prime product pp(n) >/ 19*n for every natural n >/ 7

This wlog  (for  n >/ 7)  can be read independently of wlog https://aot2025.blogspot.com/2025/09/prime-product-ppn-5n-of-primes-p-4-p-n.html while it is a continuation of that previous wlog (for  n >/ 5). ========================================== Terminology:   N  := {1 2 3 ...} -- the set of all natural numbers  P := {2  3  5  7  11  13  17 ... } -- the set of all primes  pp (x) := prod{ p in P: p \< x}. -- the product of all primes that are  \<  x ======================================================== The product of the empty set, i.e. of zero elements, is equal to 1 (by a definition). This starts the list below: pp (1) = 1 pp (2) = 2 pp (3) = 2*3 = 6 pp (4) =  pp (3) = 6 pp (5) = 2*3*5 = 30 pp (6) =  pp (5) = 30 pp (7) = 2*3*5*7 = 210 pp (10) =  pp (9) =  pp (8) =  pp (7) = 210 pp (11) = 2*3*5*7*11 = 2310 pp (12) = pp (11) = 2310 pp (13) = 2*3*5*7*11*13 = 30030...
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Prime product pp(n) >/ 5*n for every natural n >/ 5

Terminology:   N  := {1 2 3 ...} -- the set of all natural numbers  P := {2  3  5  7  11  13  17 ... } -- the set of all primes  pp (x) := prod{ p in P: p \< x}. -- the product of all primes that are  \<  x ======================================================== The product of the empty set, i.e. of zero elements, is equal to 1 (by a definition). This starts the list below: pp (1) = 1 pp (2) = 2 pp (3) = 2*3 = 6 pp (4) =  pp (3) = 6 pp (5) = 2*3*5 = 30 pp (6) =  pp (5) = 30 pp (7) = 2*3*5*7 = 210 pp (10) =  pp (9) =  pp (8) =  pp (7) = 210 pp (11) = 2*3*5*7*11 = 2310 ======================================================== Theorem    For every  n  in  N  pp (n) >/ n  n >/ 3   ==>    pp (n)  >/  n + 2  n >/ 5   ==>    pp (n)  >/  5*n Proof   A simple direct computation show...
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The arithmetics of natural numbers

 The standard way of introducing the arithmetics of natural numbers is via Peano axioms -- it is a very ascetic way. However, I'd like a comfortable way that will lead to the number theory much faster. Many people have an idea of natural numbers as a set  N = {1 2 ...}  together with operations  +  and  *  performed on natural numbers, i.e. on elements  1 2 ...  of  N .  In this wlog this idea will get a precise meaning. ========================== Asscio ( N +) ---------------- We start with a set  N , and with an associative operation binary (2-argument) operation  +.  Being an operation means that  a+b  belongs to  N   for all  a b  belonging to  N . Thus, the pair                            ( N +) forms an ascio -- it's just a very simple kind of algebra. Operation  +  is called  addition . Rem...
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2-power teams, and their leaders (AoT, nr.1)

2-power team definition ------------------------------                                     A 2-power  team  is a quadruple  A := (a  b  c  d)  of integers such that two conditions are satisfied: (i)    a  >  b  >/  c >  d  >/  0  --  a team inequality; (ii)  a^2 + d^2  =  b^2 + c^2 -- a team equation. Also the two more more equations equivalent to (ii) are called team equations: (ii')                    a^2 - b^2  =  c^2 - d^2 (ii'')                    a^2 - c^2  =  b^2 - d^2 Then  integer  a  is called the  leader  of team  A.  We'd like to know everything about the leaders (and about the teams too). Furthermore, 2-power team...
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Even, odd, and square integers (AoT, nr.0)

Notation and terminology: Different fonts represent different mathematical objects. In particular the bold font  Z   always represents the set of all (rational) integers, see below, while z z z z Z Z Z  (can your eye see the optical difference between different fonts? -- don't worry! ☺ ) they represent some other objects; furthermore any of these other letters may represent different object on different occasions as it would be stated explicitly each time. Z   =  { ... -2  -2  0  1  2 .., } -- the set of all (rational) integers; N   :=  { n \in Z : n > 0 }  =  { 1  2  3  ... } -- the set of all natural numbers (positive integers). ============================================= Introduction ---------------- Let's add the first six positive (natural) integers, and we get:          1 + 3 + 5 + 7 + 9 + 11  =   6^2 This holds for any number of the first odd numbers. Ad...
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