The determination of all prime numbers in a selected range with the n-method

PRIME NUMBER SERIES 6

The n-method is based on premise (1) and the set operations (2) to (5):

(1) 2 is the smallest and the one and only even prime number.

(2) $\mathbb{P}$ \ {2} = {p | p = 2n_p+3 ʌ n_p ∈ $\mathbb{N}_{p}}$ }

(3) $\mathbb{N}_{p}}$ = {n_p | n_p ∈ $\mathbb{N}$ ʌ n_p≠ n_km}

(4) $\mathbb{N}_{p}}$ = $\mathbb{N}}$ \ $\mathbb{N}_{km}}$

(5) $\mathbb{N}_{km}}$ = {n_km | n_km= 2k²+6k+3+m(2k+3) ʌ k, m ∈ $\mathbb{N}}$ }

PROOF of (1):

(a) Because there exists no natural number between 1 and 2, 2 can be only divisible integrally by 1 and itself. Therfore 2 is the smallest prime number.

(b) Every even number greater than 2 is given by

(2c + 4) ʌ c ∈ ℕ

This can be represented as product

2∙ (c+2) ʌ c ∈ ℕ

Thus every even number >2 is divisible integrally by 2 and therefore not a prime number.

Since no prime >2 is even, all other primes must be odd. The formula

p = 2n_p+3

yields only odd numbers, since 2n_pfor n_p ∈ ℕ is always even or 0 and an even number or 0 added with 3 yields an odd number.

PROOF of (5):

If one sets

u_km= 2n_km+3

and substitutes n_kmfrom

(5) N_km = {n_km | n_km= 2k²+6k+3+m(2k+3) ʌ k, m ∈ ℕ}

results in

u_km = 2(2k²+6k+3+m(2k+3))+3

= 4k²+12k+6+2m(2k+3)+3

= 4k²+12k+9+2m(2k+3)

u_km = (2k+3)²+2m(2k+3)

and further

u_km =(2k+3) (2k+3+2m)

u_kmis therefore the product of the odd numbers

2k+3

and

2k+3+2m

With k,m ∈ ℕ, (2k+3)(2k+3+2m) yields all possible products of odd numbers greater than 1, and hence all composite odd numbers greater than 1 that exist.

Thus yields

$\mathbb{N}_{km}}$ = {n_km | n_km= 2k²+6k+3+m(2k+3) ʌ k, m ∈ $\mathbb{N}}$ }

all n_km which are given in the formula

u_km=2n_km+3

which yield all composite odd numbers greater than 1.

PROOF of (4) $\mathbb{N}_{p}}$ = $\mathbb{N}$ \ $\mathbb{N}_{km}}$ :

The set of natural numbers $\mathbb{N}$ includes all n, which yield in the formula

U=2n+3

all odd numbers greater than 1. And all odd numbers greater than 1 can only be composite or non-composite (=prime). Since

n_km = 2k²+6k+3+m(2k+3) ʌ k, m ∈ $\mathbb{N}$

yields all n, which give all composite numbers in U=2n+3, all others must be n=n_p. Consequently

$\mathbb{N}_{p}}$ = $\mathbb{N}$ \ $\mathbb{N}_{km}}$

must be correct.

PROOF of (2) $\mathbb{P}$ \ {2} = {p | p = 2n_p+3 ʌ n_p ∈ $\mathbb{N}_{p}}$ }

and

(3) $\mathbb{N}_{p}}$ = {n_p | n_p ∈ $\mathbb{N}$ ʌ n_p≠ n_km}:

In U=2n+3, n can be only one of the two: n_km or n_p.

For the rest of the proof, suppose there is the case where n=n_km=n_p. But since n_km supplies all n, which in 2n+3 supplies all composite odd numbers, n_km cannot supply n, which supply non-composite numbers (=primes). So there must be

$\mathbb{P}$ \ {2} = {p | p = 2n_p+3 ʌ n_p ∈ $\mathbb{N}_{p}}$ }

and

$\mathbb{N}_{p}}$ = {n_p | n_p ∈ $\mathbb{N}$ ʌ n_p≠ n_km}

be correct.

What needed to be proved.

EXAMPLE 1: It shall be determined all prime numbers between 7100 and 7200 if existing:

U_min	=	7101	k	m_min	m_max
U_max	=	7199	0	1182	1198
n_min=⌈(U_min -3)/2⌉	=	3549	1	708	717
n_max=⌊(U_max -3)/2⌋	=	3598	2	504	510
k_min	=	0	3	390	395
k_max=⌊1/2*(-3+√U_max)⌋	=	40	4	318	321
m_min=f(k)=⌈(n_min-2k²-6k-3)/(2k+3)⌉			5	267	270
m_max=f(k)=⌊(n_max-2k²-6k-3)/(2k+3)⌋			6	230	232
			7	201	203
Legend:			8	178	179
U_min=smallest odd number in the selected range			9	159	160
U_max=largest odd number in the selected range			10	143	145
			11	130	131
			12	118	119
			13	108	109
			14	100	100
			15	92	92
			16	84	85
			17	78	78
			18	72	72
			19	67	67
			20	62	62
			21	57	57
			22	53	53
			23	48	48
			24	45	45
			25	41	41
			26	38	37
			27	34	34
			28	31	31
			29	28	28
			30	25	25
			31	23	22
			32	20	20
			33	17	17
			34	15	15
			35	13	12
			36	10	10
			37	8	8
			38	6	6
			39	4	3
			40	2	1

			7100<p<7200
n	n_km	n_p=n-n_km	p=2n_p+3	k	m	n_km=2k²+6k+3+m(2k+3)
3549	3549	0	3	0	1182	3549
3550		3550	7103	0	1183	3552
3551	3551	0	3	0	1184	3555
3552	3552	0	3	0	1185	3558
3553		3553	7109	0	1186	3561
3554	3554	0	3	0	1187	3564
3555	3555	0	3	0	1188	3567
3556	3556	0	3	0	1189	3570
3557	3557	0	3	0	1190	3573
3558	3558	0	3	0	1191	3576
3559		3559	7121	0	1192	3579
3560	3560	0	3	0	1193	3582
3561	3561	0	3	0	1194	3585
3562		3562	7127	0	1195	3588
3563		3563	7129	0	1196	3591
3564	3564	0	3	0	1197	3594
3565	3565	0	3	0	1198	3597
3566	3566	0	3	1	708	3551
3567	3567	0	3	1	709	3556
3568	3568	0	3	1	710	3561
3569	3569	0	3	1	711	3566
3570	3570	0	3	1	712	3571
3571	3571	0	3	1	713	3576
3572	3572	0	3	1	714	3581
3573	3573	0	3	1	715	3586
3574		3574	7151	1	716	3591
3575	3575	0	3	1	717	3596
3576	3576	0	3	2	504	3551
3577	3577	0	3	2	505	3558
3578		3578	7159	2	506	3565
3579	3579	0	3	2	507	3572
3580	3580	0	3	2	508	3579
3581	3581	0	3	2	509	3586
3582	3582	0	3	2	510	3593
3583	3583	0	3	3	390	3549
3584	3584	0	3	3	391	3558
3585	3585	0	3	3	392	3567
3586	3586	0	3	3	393	3576
3587		3587	7177	3	394	3585
3588	3588	0	3	3	395	3594
3589	3589	0	3	4	318	3557
3590	3590	0	3	4	319	3568
3591	3591	0	3	4	320	3579
3592		3592	7187	4	321	3590
3593	3593	0	3	5	267	3554
3594	3594	0	3	5	268	3567
3595		3595	7193	5	269	3580
3596	3596	0	3	5	270	3593
3597	3597	0	3	6	230	3561
3598	3598	0	3	6	231	3576
				6	232	3591
				7	201	3560
				7	202	3577
				7	203	3594
				8	178	3561
				8	179	3580
				9	159	3558
				9	160	3579
				10	143	3552
				10	144	3575
				10	145	3598
				11	130	3561
				11	131	3586
				12	118	3549
				12	119	3576
				13	108	3551
				13	109	3580
				14	100	3579
				15	92	3579
				16	84	3551
				16	85	3586
				17	78	3569
				18	72	3567
				19	67	3586
				20	62	3589
				21	57	3576
				22	53	3594
				23	48	3551
				24	45	3594
				25	41	3576
				26	38	3601
				27	34	3561
				28	31	3568
				29	28	3567
				30	25	3558
				31	23	3606
				32	20	3583
				33	17	3552
				34	15	3584
				35	13	3612
				36	10	3561
				37	8	3579
				38	6	3593
				39	4	3603
				40	2	3609

Ergo: The procedure supplies completely all prime numbers (bold type) in the choosen example

between 7100 and 7200.

EXAMPLE 2: It shall be determined all prime numbers >2 and <1000:

U_min	=	3	k	m_min	m_max
U_max	=	999	0	0	165
n_min=⌈(U_min -3)/2⌉	=	0	1	0	97
n_max=⌊(U_max -3)/2⌋	=	498	2	0	67
k_min	=	0	3	0	51
k_max=⌊1/2*(-3+√U_max)⌋	=	14	4	0	39
m_min=f(k)=⌈(n_min-2k²-6k-3)/(2k+3)⌉	=	0	5	0	31
m_max=f(k)=⌊(n_max-2k²-6k-3)/(2k+3)⌋			6	0	25
			7	0	20
Legend:			8	0	16
U_min=smallest odd number in the selected range			9	0	13
U_max=largest odd number in the selected range			10	0	10
			11	0	7
			12	0	5
			13	0	2
			14	0	0

			2<p<1000
n	n_km	n_p=n-n_km	p=2n_p+3	k	m	n_t _km=2k²+6k+3+m(2k+3)
0		0	3	0	0	3
1		1	5	0	1	6
2		2	7	0	2	9
3	3	0	3	0	3	12
4		4	11	0	4	15
5		5	13	0	5	18
6	6	0	3	0	6	21
7		7	17	0	7	24
8		8	19	0	8	27
9	9	0	3	0	9	30
10		10	23	0	10	33
11	11	0	3	0	11	36
12	12	0	3	0	12	39
13		13	29	0	13	42
14		14	31	0	14	45
15	15	0	3	0	15	48
16	16	0	3	0	16	51
17		17	37	0	17	54
18	18	0	3	0	18	57
19		19	41	0	19	60
20		20	43	0	20	63
21	21	0	3	0	21	66
22		22	47	0	22	69
23	23	0	3	0	23	72
24	24	0	3	0	24	75
25		25	53	0	25	78
26	26	0	3	0	26	81
27	27	0	3	0	27	84
28		28	59	0	28	87
29		29	61	0	29	90
30	30	0	3	0	30	93
31	31	0	3	0	31	96
32		32	67	0	32	99
33	33	0	3	0	33	102
34		34	71	0	34	105
35		35	73	0	35	108
36	36	0	3	0	36	111
37	37	0	3	0	37	114
38		38	79	0	38	117
39	39	0	3	0	39	120
40		40	83	0	40	123
41	41	0	3	0	41	126
42	42	0	3	0	42	129
43		43	89	0	43	132
44	44	0	3	0	44	135
45	45	0	3	0	45	138
46	46	0	3	0	46	141
47		47	97	0	47	144
48	48	0	3	0	48	147
49		49	101	0	49	150
50		50	103	0	50	153
51	51	0	3	0	51	156
52		52	107	0	52	159
53		53	109	0	53	162
54	54	0	3	0	54	165
55		55	113	0	55	168
56	56	0	3	0	56	171
57	57	0	3	0	57	174
58	58	0	3	0	58	177
59	59	0	3	0	59	180
60	60	0	3	0	60	183
61	61	0	3	0	61	186
62		62	127	0	62	189
63	63	0	3	0	63	192
64		64	131	0	64	195
65	65	0	3	0	65	198
66	66	0	3	0	66	201
67		67	137	0	67	204
68		68	139	0	68	207
69	69	0	3	0	69	210
70	70	0	3	0	70	213
71	71	0	3	0	71	216
72	72	0	3	0	72	219
73		73	149	0	73	222
74		74	151	0	74	225
75	75	0	3	0	75	228
76	76	0	3	0	76	231
77		77	157	0	77	234
78	78	0	3	0	78	237
79	79	0	3	0	79	240
80		80	163	0	80	243
81	81	0	3	0	81	246
82		82	167	0	82	249
83	83	0	3	0	83	252
84	84	0	3	0	84	255
85		85	173	0	85	258
86	86	0	3	0	86	261
87	87	0	3	0	87	264
88		88	179	0	88	267
89		89	181	0	89	270
90	90	0	3	0	90	273
91	91	0	3	0	91	276
92	92	0	3	0	92	279
93	93	0	3	0	93	282
94		94	191	0	94	285
95		95	193	0	95	288
96	96	0	3	0	96	291
97		97	197	0	97	294
98		98	199	0	98	297
99	99	0	3	0	99	300
100	100	0	3	0	100	303
101	101	0	3	0	101	306
102	102	0	3	0	102	309
103	103	0	3	0	103	312
104		104	211	0	104	315
105	105	0	3	0	105	318
106	106	0	3	0	106	321
107	107	0	3	0	107	324
108	108	0	3	0	108	327
109	109	0	3	0	109	330
110		110	223	0	110	333
111	111	0	3	0	111	336
112		112	227	0	112	339
113		113	229	0	113	342
114	114	0	3	0	114	345
115		115	233	0	115	348
116	116	0	3	0	116	351
117	117	0	3	0	117	354
118		118	239	0	118	357
119		119	241	0	119	360
120	120	0	3	0	120	363
121	121	0	3	0	121	366
122	122	0	3	0	122	369
123	123	0	3	0	123	372
124		124	251	0	124	375
125	125	0	3	0	125	378
126	126	0	3	0	126	381
127		127	257	0	127	384
128	128	0	3	0	128	387
129	129	0	3	0	129	390
130		130	263	0	130	393
131	131	0	3	0	131	396
132	132	0	3	0	132	399
133		133	269	0	133	402
134		134	271	0	134	405
135	135	0	3	0	135	408
136	136	0	3	0	136	411
137		137	277	0	137	414
138	138	0	3	0	138	417
139		139	281	0	139	420
140		140	283	0	140	423
141	141	0	3	0	141	426
142	142	0	3	0	142	429
143	143	0	3	0	143	432
144	144	0	3	0	144	435
145		145	293	0	145	438
146	146	0	3	0	146	441
147	147	0	3	0	147	444
148	148	0	3	0	148	447
149	149	0	3	0	149	450
150	150	0	3	0	150	453
151	151	0	3	0	151	456
152		152	307	0	152	459
153	153	0	3	0	153	462
154		154	311	0	154	465
155		155	313	0	155	468
156	156	0	3	0	156	471
157		157	317	0	157	474
158	158	0	3	0	158	477
159	159	0	3	0	159	480
160	160	0	3	0	160	483
161	161	0	3	0	161	486
162	162	0	3	0	162	489
163	163	0	3	0	163	492
164		164	331	0	164	495
165	165	0	3	0	165	498
166	166	0	3	1	0	11
167		167	337	1	1	16
168	168	0	3	1	2	21
169	169	0	3	1	3	26
170	170	0	3	1	4	31
171	171	0	3	1	5	36
172		172	347	1	6	41
173		173	349	1	7	46
174	174	0	3	1	8	51
175		175	353	1	9	56
176	176	0	3	1	10	61
177	177	0	3	1	11	66
178		178	359	1	12	71
179	179	0	3	1	13	76
180	180	0	3	1	14	81
181	181	0	3	1	15	86
182		182	367	1	16	91
183	183	0	3	1	17	96
184	184	0	3	1	18	101
185		185	373	1	19	106
186	186	0	3	1	20	111
187	187	0	3	1	21	116
188		188	379	1	22	121
189	189	0	3	1	23	126
190		190	383	1	24	131
191	191	0	3	1	25	136
192	192	0	3	1	26	141
193		193	389	1	27	146
194	194	0	3	1	28	151
195	195	0	3	1	29	156
196	196	0	3	1	30	161
197		197	397	1	31	166
198	198	0	3	1	32	171
199		199	401	1	33	176
200	200	0	3	1	34	181
201	201	0	3	1	35	186
202	202	0	3	1	36	191
203		203	409	1	37	196
204	204	0	3	1	38	201
205	205	0	3	1	39	206
206	206	0	3	1	40	211
207	207	0	3	1	41	216
208		208	419	1	42	221
209		209	421	1	43	226
210	210	0	3	1	44	231
211	211	0	3	1	45	236
212	212	0	3	1	46	241
213	213	0	3	1	47	246
214		214	431	1	48	251
215		215	433	1	49	256
216	216	0	3	1	50	261
217	217	0	3	1	51	266
218		218	439	1	52	271
219	219	0	3	1	53	276
220		220	443	1	54	281
221	221	0	3	1	55	286
222	222	0	3	1	56	291
223		223	449	1	57	296
224	224	0	3	1	58	301
225	225	0	3	1	59	306
226	226	0	3	1	60	311
227		227	457	1	61	316
228	228	0	3	1	62	321
229		229	461	1	63	326
230		230	463	1	64	331
231	231	0	3	1	65	336
232		232	467	1	66	341
233	233	0	3	1	67	346
234	234	0	3	1	68	351
235	235	0	3	1	69	356
236	236	0	3	1	70	361
237	237	0	3	1	71	366
238		238	479	1	72	371
239	239	0	3	1	73	376
240	240	0	3	1	74	381
241	241	0	3	1	75	386
242		242	487	1	76	391
243	243	0	3	1	77	396
244		244	491	1	78	401
245	245	0	3	1	79	406
246	246	0	3	1	80	411
247	247	0	3	1	81	416
248		248	499	1	82	421
249	249	0	3	1	83	426
250		250	503	1	84	431
251	251	0	3	1	85	436
252	252	0	3	1	86	441
253		253	509	1	87	446
254	254	0	3	1	88	451
255	255	0	3	1	89	456
256	256	0	3	1	90	461
257	257	0	3	1	91	466
258	258	0	3	1	92	471
259		259	521	1	93	476
260		260	523	1	94	481
261	261	0	3	1	95	486
262	262	0	3	1	96	491
263	263	0	3	1	97	496
264	264	0	3	2	0	23
265	265	0	3	2	1	30
266	266	0	3	2	2	37
267	267	0	3	2	3	44
268	268	0	3	2	4	51
269		269	541	2	5	58
270	270	0	3	2	6	65
271	271	0	3	2	7	72
272		272	547	2	8	79
273	273	0	3	2	9	86
274	274	0	3	2	10	93
275	275	0	3	2	11	100
276	276	0	3	2	12	107
277		277	557	2	13	114
278	278	0	3	2	14	121
279	279	0	3	2	15	128
280		280	563	2	16	135
281	281	0	3	2	17	142
282	282	0	3	2	18	149
283		283	569	2	19	156
284		284	571	2	20	163
285	285	0	3	2	21	170
286	286	0	3	2	22	177
287		287	577	2	23	184
288	288	0	3	2	24	191
289	289	0	3	2	25	198
290	290	0	3	2	26	205
291	291	0	3	2	27	212
292		292	587	2	28	219
293	293	0	3	2	29	226
294	294	0	3	2	30	233
295		295	593	2	31	240
296	296	0	3	2	32	247
297	297	0	3	2	33	254
298		298	599	2	34	261
299		299	601	2	35	268
300	300	0	3	2	36	275
301	301	0	3	2	37	282
302		302	607	2	38	289
303	303	0	3	2	39	296
304	304	0	3	2	40	303
305		305	613	2	41	310
306	306	0	3	2	42	317
307		307	617	2	43	324
308		308	619	2	44	331
309	309	0	3	2	45	338
310	310	0	3	2	46	345
311	311	0	3	2	47	352
312	312	0	3	2	48	359
313	313	0	3	2	49	366
314		314	631	2	50	373
315	315	0	3	2	51	380
316	316	0	3	2	52	387
317	317	0	3	2	53	394
318	318	0	3	2	54	401
319		319	641	2	55	408
320		320	643	2	56	415
321	321	0	3	2	57	422
322		322	647	2	58	429
323	323	0	3	2	59	436
324	324	0	3	2	60	443
325		325	653	2	61	450
326	326	0	3	2	62	457
327	327	0	3	2	63	464
328		328	659	2	64	471
329		329	661	2	65	478
330	330	0	3	2	66	485
331	331	0	3	2	67	492
332	332	0	3	3	0	39
333	333	0	3	3	1	48
334	334	0	3	3	2	57
335		335	673	3	3	66
336	336	0	3	3	4	75
337		337	677	3	5	84
338	338	0	3	3	6	93
339	339	0	3	3	7	102
340		340	683	3	8	111
341	341	0	3	3	9	120
342	342	0	3	3	10	129
343	343	0	3	3	11	138
344		344	691	3	12	147
345	345	0	3	3	13	156
346	346	0	3	3	14	165
347	347	0	3	3	15	174
348	348	0	3	3	16	183
349		349	701	3	17	192
350	350	0	3	3	18	201
351	351	0	3	3	19	210
352	352	0	3	3	20	219
353		353	709	3	21	228
354	354	0	3	3	22	237
355	355	0	3	3	23	246
356	356	0	3	3	24	255
357	357	0	3	3	25	264
358		358	719	3	26	273
359	359	0	3	3	27	282
360	360	0	3	3	28	291
361	361	0	3	3	29	300
362		362	727	3	30	309
363	363	0	3	3	31	318
364	364	0	3	3	32	327
365		365	733	3	33	336
366	366	0	3	3	34	345
367	367	0	3	3	35	354
368		368	739	3	36	363
369	369	0	3	3	37	372
370		370	743	3	38	381
371	371	0	3	3	39	390
372	372	0	3	3	40	399
373	373	0	3	3	41	408
374		374	751	3	42	417
375	375	0	3	3	43	426
376	376	0	3	3	44	435
377		377	757	3	45	444
378	378	0	3	3	46	453
379		379	761	3	47	462
380	380	0	3	3	48	471
381	381	0	3	3	49	480
382	382	0	3	3	50	489
383		383	769	3	51	498
384	384	0	3	4	0	59
385		385	773	4	1	70
386	386	0	3	4	2	81
387	387	0	3	4	3	92
388	388	0	3	4	4	103
389	389	0	3	4	5	114
390	390	0	3	4	6	125
391	391	0	3	4	7	136
392		392	787	4	8	147
393	393	0	3	4	9	158
394	394	0	3	4	10	169
395	395	0	3	4	11	180
396	396	0	3	4	12	191
397		397	797	4	13	202
398	398	0	3	4	14	213
399	399	0	3	4	15	224
400	400	0	3	4	16	235
401	401	0	3	4	17	246
402	402	0	3	4	18	257
403		403	809	4	19	268
404		404	811	4	20	279
405	405	0	3	4	21	290
406	406	0	3	4	22	301
407	407	0	3	4	23	312
408	408	0	3	4	24	323
409		409	821	4	25	334
410		410	823	4	26	345
411	411	0	3	4	27	356
412		412	827	4	28	367
413		413	829	4	29	378
414	414	0	3	4	30	389
415	415	0	3	4	31	400
416	416	0	3	4	32	411
417	417	0	3	4	33	422
418		418	839	4	34	433
419	419	0	3	4	35	444
420	420	0	3	4	36	455
421	421	0	3	4	37	466
422	422	0	3	4	38	477
423	423	0	3	4	39	488
424	424	0	3	5	0	83
425		425	853	5	1	96
426	426	0	3	5	2	109
427		427	857	5	3	122
428		428	859	5	4	135
429	429	0	3	5	5	148
430		430	863	5	6	161
431	431	0	3	5	7	174
432	432	0	3	5	8	187
433	433	0	3	5	9	200
434	434	0	3	5	10	213
435	435	0	3	5	11	226
436	436	0	3	5	12	239
437		437	877	5	13	252
438	438	0	3	5	14	265
439		439	881	5	15	278
440		440	883	5	16	291
441	441	0	3	5	17	304
442		442	887	5	18	317
443	443	0	3	5	19	330
444	444	0	3	5	20	343
445	445	0	3	5	21	356
446	446	0	3	5	22	369
447	447	0	3	5	23	382
448	448	0	3	5	24	395
449	449	0	3	5	25	408
450	450	0	3	5	26	421
451	451	0	3	5	27	434
452		452	907	5	28	447
453	453	0	3	5	29	460
454		454	911	5	30	473
455	455	0	3	5	31	486
456	456	0	3	6	0	111
457	457	0	3	6	1	126
458		458	919	6	2	141
459	459	0	3	6	3	156
460	460	0	3	6	4	171
461	461	0	3	6	5	186
462	462	0	3	6	6	201
463		463	929	6	7	216
464	464	0	3	6	8	231
465	465	0	3	6	9	246
466	466	0	3	6	10	261
467		467	937	6	11	276
468	468	0	3	6	12	291
469		469	941	6	13	306
470	470	0	3	6	14	321
471	471	0	3	6	15	336
472		472	947	6	16	351
473	473	0	3	6	17	366
474	474	0	3	6	18	381
475		475	953	6	19	396
476	476	0	3	6	20	411
477	477	0	3	6	21	426
478	478	0	3	6	22	441
479	479	0	3	6	23	456
480	480	0	3	6	24	471
481	481	0	3	6	25	486
482		482	967	7	0	143
483	483	0	3	7	1	160
484		484	971	7	2	177
485	485	0	3	7	3	194
486	486	0	3	7	4	211
487		487	977	7	5	228
488	488	0	3	7	6	245
489	489	0	3	7	7	262
490		490	983	7	8	279
491	491	0	3	7	9	296
492	492	0	3	7	10	313
493	493	0	3	7	11	330
494		494	991	7	12	347
495	495	0	3	7	13	364
496	496	0	3	7	14	381
497		497	997	7	15	398
498	498	0	3	7	16	415
				7	17	432
				7	18	449
				7	19	466
				7	20	483
				8	0	179
				8	1	198
				8	2	217
				8	3	236
				8	4	255
				8	5	274
				8	6	293
				8	7	312
				8	8	331
				8	9	350
				8	10	369
				8	11	388
				8	12	407
				8	13	426
				8	14	445
				8	15	464
				8	16	483
				9	0	219
				9	1	240
				9	2	261
				9	3	282
				9	4	303
				9	5	324
				9	6	345
				9	7	366
				9	8	387
				9	9	408
				9	10	429
				9	11	450
				9	12	471
				9	13	492
				10	0	263
				10	1	286
				10	2	309
				10	3	332
				10	4	355
				10	5	378
				10	6	401
				10	7	424
				10	8	447
				10	9	470
				10	10	493
				11	0	311
				11	1	336
				11	2	361
				11	3	386
				11	4	411
				11	5	436
				11	6	461
				11	7	486
				12	0	363
				12	1	390
				12	2	417
				12	3	444
				12	4	471
				12	5	498
				13	0	419
				13	1	448
				13	2	477
				14	0	479

Ergo: The procedure supplies completely all prime numbers (bold type) in the choosen example

>2 and <1000.

Munich, 7 August 2019

Gottfried Färberböck

Language:

Language:

THE PRIME NUMBER SERIES

The determination of all prime numbers in a selected range with the n-method