On Counting [ Candidates-for-Primes modulo 30 ] up to [ N = 1543 ]

by Frank C. Fung ( 1st published in October, 2012. )

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Section V - The Power-Roots of [ N ]

Summary for the Section :

• In this Section , we list-out the power-roots of [ N = 1543 ] ,
  • for our reference and usage in later Sections .

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The Power-Roots of [ N ] :

• We then list-out the power-roots of [ N ] in this table below :

  Values for the [ i-th root ] of [ N ]		* * *	Values for the Powers of [ X ]
  Description	Value of i-th -root ( X )		X^1	X^2	X^3	X^4	X^5	X^6	X^7	X^8	X^9	X^10	X^11
  SELF	1543.0000	***	1543
  Square-Root	39.2810	***	39.28	1543
  Cube-Root	11.5555	***	11.56	133.53	1543
  4th-Root	6.2675	***	6.27	39.28	246.19	1543
  5th-Root	4.3418	***	4.34	18.85	81.85	355.38	1543
  6th-Root	3.3993	***	3.40	11.56	39.28	133.53	453.91	1543
  7th-Root	2.8542	***	2.85	8.15	23.25	66.36	189.41	540.61	1543
  8th-Root	2.5035	***	2.50	6.27	15.69	39.28	98.34	246.19	616.34	1543
  9th-Root	2.2608	***	2.26	5.11	11.56	26.12	59.06	133.53	301.88	682.50	1543
  10th-Root	2.0837	***	2.08	4.34	9.05	18.85	39.28	81.85	170.55	355.38	740.51	1543
  11th-Root	1.9492	***	1.95	3.80	7.41	14.43	28.14	54.84	106.90	208.36	406.13	791.62	1543
  ***********	***********	***	******	******	******	******	******	******	******	******	******	******	******

  And we simply note here that :

  • the cube-root of [ 1543 ] is roughly [ 11.56 ] , and
  • the 4th-root of [ 1543 ] is roughly [ 6.27 ] .

• As such :
  • the { 4th-root of [ N = 1543 ] } at [ 6.27 ] is smaller than smallest prime-number under consideration here :
    • namely : the prime-number [ 7 ] .

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go to the next section : Section VI - Counting { Category II Candidates } up to [ N ]

go to the last section : Section IV - The Powers-of-[ 7 ] in relation to [ N ]

return to the Counting via [ modulo 30 ] HomePage

Original dated 2012-10-08