Primes p,q : (k+1)+(k+2)+(k+3)+..+(k+p) = (n+1)+(n+2)+(n+3)+…+(n+q) = x^2

 

Prime   p   expressed as

p^2 \; = \; (k+1) \; + \; (k+2) \; + \; (k+3) \; + \; ... \; + \; (k+p)

  3^2 \; = \; (1+1) \; + \; (1+2) \; + \; (1+3)
  5^2 \; = \; (2+1) \; + \; (2+2) \; + \; (2+3) \; + \; (2+4) \; + \; (2+5)
  7^2 \; = \; (3+1) \; + \; (3+2) \; + \; (3+3) \; + \; ... \; + \; (3+7)

11^2 \; = \; (5+1) \; + \; (5+2) \; + \; (5+3) \; + \; ... \; + \; (5+11)
13^2 \; = \; (6+1) \; + \; (6+2) \; + \; (6+3) \; + \; ... \; + \; (6+13)
17^2 \; = \; (8+1) \; + \; (8+2) \; + \; (8+3) \; + \; ... \; + \; (8+17)
19^2 \; = \; (9+1) \; + \; (9+2) \; + \; (9+3) \; + \; ... \; + \; (9+19)

23^2 \; = \; (11+1) \; + \; (11+2) \; + \; (11+3) \; + \; ... \; + \; (11+23)
29^2 \; = \; (14+1) \; + \; (14+2) \; + \; (14+3) \; + \; ... \; + \; (14+29)
31^2 \; = \; (15+1) \; + \; (15+2) \; + \; (15+3) \; + \; ... \; + \; (15+31)
37^2 \; = \; (18+1) \; + \; (18+2) \; + \; (18+3) \; + \; ... \; + \; (18+37)
41^2 \; = \; (20+1) \; + \; (20+2) \; + \; (20+3) \; + \; ... \; + \; (20+41)
43^2 \; = \; (21+1) \; + \; (21+2) \; + \; (21+3) \; + \; ... \; + \; (21+43)
47^2 \; = \; (23+1) \; + \; (23+2) \; + \; (23+3) \; + \; ... \; + \; (23+47)
53^2 \; = \; (26+1) \; + \; (26+2) \; + \; (26+3) \; + \; ... \; + \; (26+53)
59^2 \; = \; (29+1) \; + \; (29+2) \; + \; (29+3) \; + \; ... \; + \; (29+59)
61^2 \; = \; (30+1) \; + \; (30+2) \; + \; (30+3) \; + \; ... \; + \; (30+61)
67^2 \; = \; (33+1) \; + \; (33+2) \; + \; (33+3) \; + \; ... \; + \; (33+67)
71^2 \; = \; (35+1) \; + \; (35+2) \; + \; (35+3) \; + \; ... \; + \; (35+71)
73^2 \; = \; (36+1) \; + \; (36+2) \; + \; (36+3) \; + \; ... \; + \; (36+73)
79^2 \; = \; (39+1) \; + \; (39+2) \; + \; (39+3) \; + \; ... \; + \; (39+79)
83^2 \; = \; (41+1) \; + \; (41+2) \; + \; (41+3) \; + \; ... \; + \; (41+83)
89^2 \; = \; (44+1) \; + \; (44+2) \; + \; (44+3) \; + \; ... \; + \; (44+89)
97^2 \; = \; (48+1) \; + \; (48+2) \; + \; (48+3) \; + \; ... \; + \; (48+97)

 
 

with prime   q > 3   and   p = 3

If we set   (k+1)+(k+2)+(k+3) \; = \; (n+1)+(n+2)+(n+3)+...+(n+q) \; = \; x^2

explain why   x = 3 \, q

15^2 = (3\times 5)^2
= (73+1)+(73+2)+(73+3)
= (42+1)+(42+2)+(42+3)+(42+4)+(42+5)

21^2 = (3\times 7)^2
= (145+1)+(145+2)+(145+3)
= (59+1)+(59+2)+(59+3)+...+(59+7)

33^2 = (3\times 11)^2
= (361+1)+(361+2)+(361+3)
= (93+1)+(93+2)+(93+3)+...+(93+11)

39^2 = (3\times 13)^2
= (505+1)+(505+2)+(505+3)
= (110+1)+(110+2)+(110+3)+...+(110+13)

51^2 = (3\times 17)^2
= (865+1)+(865+2)+(865+3)
= (144+1)+(144+2)+(144+3)+...+(144+17)

57^2 = (3\times 19)^2
= (1081+1)+(1081+2)+(1081+3)
= (161+1)+(161+2)+(161+3)+...+(161+19)

 

now for   p > 3

(k+1)+(k+2)+(k+3)+...+(k+p)
= \; (n+1)+(n+2)+(n+3)+...+(n+q) \; = \; x^2

p = 5

35^2 = (5\times 7)^2
= (242+1)+(242+2)+(242+3)+(242+4)+(242+5)
= (171+1)+(171+2)+(171+3)+...+(171+7)

55^2 = (5\times 11)^2
= (602+1)+(602+2)+(602+3)+(602+4)+(602+5)
= (269+1)+(269+2)+(269+3)+...+(269+11)

p = 7

77^2 = (7\times 11)^2
= (843+1)+(843+2)+(843+3)+...+(843+7)
= (533+1)+(533+2)+(533+3)+...+(533+11)

91^2 = (7\times 13)^2
= (1179+1)+(1179+2)+(1179+3)+...+(1179+7)
= (630+1)+(630+2)+(630+3)+...+(630+13)

 
 

Can we generalize these results?

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Prime Numbers and tagged . Bookmark the permalink.

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