pier francesco roggero, michele nardelli, francesco di noto - "there is no secure test for...

7/27/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - "There is No Secure Test for Primality"

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THERE IS NO SECURE TEST FOR PRIMALITY

Ing. Pier Francesco Roggero, Dott. Michele Nardelli, Francesco Di Noto

Abstract

In this paper we examine in detail a completely new test for the primality of

prime number and how all these tests behave in the mystery of prime

numbers.


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Index:

1. PRIMALITY TEST ............................................................................................................................. 31.1 SUM OF THE DIGITS OF NUMBER N .............................................................................. 41.2 PRODUCT OF THE DIGITS OF THE NUMBER N ........................................................... 51.3 ANALYSIS OF PAIRS S e P ................................................................................................ 71.4 FURTHER INVESTIGATIONS ........................................................................................... 91.5 CONNECTION WITH PERFECT NUMBERS ..................................................................14

2. CONCLUSIONS................................................................................................................................163. ANALYSIS OF 1229 PRIMES LESS THAN 10000........................................................................184. REFERENCES ..................................................................................................................................67


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1. PRIMALITY TEST

Aprimality test is an algorithm that allows to decide whether a given number is prime

or not.

We try to build one.

Given a number N with the last digit ends in 1, 3, 7 or 9 we add up its digits in modulo

9 (as is done with the proof of 9) and then multiply each digits in modulo 9.


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1.1 SUM OF THE DIGITS OF NUMBER N

As the number N can also be written in polynomial form

N = an10n+ + a110

1+ a010

0= ai10

i

We have

airepresents the sum of all its digits.

Now the sum of its digits modulo 9 represents the REMAINDER of the division by 9,and then we can write:

S ai mod 9

From the sum are automatically removed 9s and the digits that give as the sum 9 or

multiple, in that its congruence is 0, resulting in a neutral element in the sum that may

well be omitted.

If the sum S is equal to 0, 3 or 6 the number is divisible by 3.


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1.2PRODUCT OF THE DIGITS OF THE NUMBER N

Similarly, here we have:

N = an10n+ + a110

1+ a010

0= ai10

i

The product of all its digits is given by:

airepresents the product of all its digits.

Now the product of its digits in module 9 represents, also in this case, the remainder ofthe division by 9, and then we can write:

P ai mod 9

In the product if it appears in the digits a 0 or a 9 or digits that give the product as 9

(3*3, 3*3*3, ) or multiple (6 * 3 6 * 6, ...), it is useless to continue to multiply hisfigures because P = 0


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From the product P can derive the following interesting table:

TAB 1:

P N

Last digit of

the prime

number

0At least 1 digit of N is 0 or 9 or 2 digits or more of

N are a multiple of 9(3*3, 6*3, 3*3*6)1, 3, 7, 9

3, 6 Only 1 digit of N is 3 or 6 1, 3, 7

1, 4, 7 At least 1 digit of N is 1, 4, 7 1, 7

2, 5, 8 At least 1 digit of N is 2, 5, 8 1, 7

Note:

You cannot have a prime number N with last digit equal to 9 if its P = 3 or 6, because ifthe last digit is 9 P values 0.

If P = 1, 2, 4, 5, 7 or 8, the last digit of the first number N is 1 or 7, because in this case

in the digits of N never appear either 0, 3, 6 or 9.

If P = 3 or 6 in the prime number N appears ONLY ONE digit that is 3 or 6.

All odd numbers N of the form 6k + 1, potentially prime numbers, or multiples orpowers of primes, which contain a 0, a 9, two 3's, a 6 and a 3, not necessarily

consecutive, have P = 0, since multiple 9, because 3 * 3 = 9, 6 * 3 = 18 multiple of 9,

while a 0 cancels the product of the digits of N.


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1.3ANALYSIS OF PAIRS S e P

Analyzing the S and P pairs of 1229 prime numbers less than 10,000, some pairs are

not present.

More specifically, the following 8 pairs S and P are NOT present:

S P1 1

5 1

7 1

7 2

1 4

2 4

4 4

4 7

In total there are 9 x 9 = 81 pairs, where 27 have S = S = 0, 3 and 6, which are multiples

of 3. There remain then 54 but ONLY 8 are not eligible to give prime numbers less than10,000.

Thus, at least for N odd numbers lower than 10 000, such pairs are good only for a

deterministic test of "non primality".


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But are we sure that continuing in our analysis also these 8 pairs do not emerge withprime numbers greater than 10,000?

In fact here are some examples of prime numbers greater than 10,000 that "cover" the

remaining pairs of S and P.

TAB. 2

Primenumber S P

11251 1 1

11287 1 4

111827 2 4

12451 4 4

11281 4 7

11111 5 1

12877 7 1111121 7 2

For pairs S and P (2,4) e (7,2) should even go to the prime numbers greater than

100,000

These pairs are veryrare but are present.

The test is probabilistic but, like all other similar tests, inefficient to have certainty.


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1.4FURTHER INVESTIGATIONS

Prior to the following table for the first 1229 prime numbers up to 10 000, ie, (10 000)

= 1229, we make some example with primes of the form 6k + 1

Table 6k +1 to 25, except for the initial number 1, to understand the general trend of S

and P, to extending it to larger numbers in case of any possible regularity

In red numbers multiple of 3, which then will be eliminated, since they are never primenumbers, except for the initial 3.

In blue the numbers of the form 6k + 1 which include prime numbers, and their products

and powers, that our test should not obviously give as prime, in black all other even

numbers and not prime, except for the initial 2 in the first column.

6k 4S P

6k-3S P

6k-2S P 6k-1S P

6kS P 6k+1S P

2

2 2

3

3 3

4

4 455 5

6

6 677 7

8

8 8

9

9 9

10

1 0112 1

12

3 2134 3

14

5 4

15

6 5

16

7 6178 7

18

9 8191 9

202 0 213 2 224 4 235 6 246 8 25 compos.7 1

26

8 3

27

9 0

28

1 7292 0

30

3 0314 3

32

5 6

33

6 0

34

7 3

35 compos.

8 6

36

9 0371 3


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Now we have the following sums S and products P, for each of the six numeric forms,

which contain all the numbers except the first initial of the form 6 * 0 +1 = 0 +1:

6k -2

S P

6k-3

S P

6k-4

S P6k-5S P

6k

S P6k+1S P

2 2 3 3 4 4 5 5 6 6 7 7

8 8 9 9 1 0 2 1 3 2 4 3

5 4 6 5 7 6 8 7 9 8 1 9

2 0 3 2 4 4 5 6 6 8 7 18 3 9 0 1 7 2 0 3 0 4 3

5 6 6 0 7 3 8 6 9 0 1 3

For our purposes only affect the columns in blue in which we note that the digits S are

repeated with the sequence 5 2 8 in column 6k -1 e 7 4 1 in column 6k + 1,while for the products P such repetition does not exist or is not known, since a table

should be much longer, what we might see later.

The cyclicity of S is due to the fact that if a number N is added to a multiple k 'of 9,with k = k*1,5 where k is the k of6k + 1 returns to the original numerical root, for the

prime numbers, we are interested in this work, we have, starting from 5, 5 + 18 = 23,

with S = 2 +3 = 5 original, etc.

3+6 0 2*9 =18 = three lines, which is why every three lines repeats the same value of S.

For the products P it is more difficult.For primes omirp, and those swappable, they all have the same pair, S and P, being the

sum and produced two commutative operations.

Eg. omirp numbers 37 and 73, with the same root number

S = 3 + 7 =7 + 3 = 10 = 1 + 0 =1 eP = 3*7= 7*3 =21 = 2+1=3

We also note, and it is logical that in the column 6k + 1 S increased by 2 compared to

the S in the column 6k -1, being essendo 6k + 1 (6k -1) = 6k +1 - 6k +1 =1+ 1 = 2,

and taking into account the form if S +2 is two digits, eg. 8 in the third and sixth row, 8

+2 = 10 and 1 + 0 = 1, in fact 8 + 2 (mod 9) = 1.


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Table of prime numbers up to 100 form 6 k -1 and relative values of S and P

Prime number of form6k -1

S cyclic: 8 5 2, 8 5 2 P not cyclic

5 5 5

11 2 1

17 8 7

23 5 629 2 9

35 composite, last digit 5 8 6

41 5 4

47 2 1

53 8 6

59 5 9


71 8 7

77 composite = 7*11 5 483 2 6

89 8 9


97 2 9

..

We note, however, that some pairs of S and P, for example 8 and 6 are both prime

numbers (53), which for composite numbers, (35), as well as 5 e 9 for the first prime

number 59 that for the composite 95 .Therefore, under this aspect, the prime numbers are not privileged by particular pairs of

S and P.


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The same goes for prime numbers of the form 6k + 1

Prime number of form6k +1

S cyclic: 7 4 1, 7 4 1 P not cyclic

7 7 7

13 4 3

19 1 9

25 composite, last digit 5 7 131 4 3

37 1 3

43 7 3

49 composite = 7*7 4 9


61 7 6

67 4 6

73 1 3

79 7 985 composite, last digit 5 4 4

91 composite =7*13 1 9

97 7 6

Also here we have obviously the same phenomenon, but with the pair 1 9, for both theprime number 19, for both the composite number 91.

But the following table, the result of an algorithm, shows all the 1229 prime numbers up

to 10000, with its pairs S and P, where we can observe the pair 8 and 6 occurs for someof them.

Here, however, are obviously not present the composite numbers with pair 8 and 6, as

we have seen for the composite number 35 (8, 6) for the prime number 53 (permutation

of 35), 467, 647 (permutation of 467), 2861, 5651, while the pair (8, 0) is present for a

large number of prime numbers.

Then each pair would have its frequency, which ranges from a minimum of attendance

(zero prime numbers for the two pairs (2, 4), (7. 2) and high, as for example the couple


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above (8, 0), there are already 16 appearances up to 1000 as the pair (8, 6) is presentonly 5 times up to 10 000.


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1.5 CONNECTION WITH PERFECT NUMBERS

The mathematical Sicilian Filippo Giordano noted that all perfect numbers, except for

the initial 6, have as the sum S = 1, for example, to 28, S = 2+8 = 10 = 1+ 0 =1, per

496, S = 4+9+6 = 19 = 1+ 9 = 10 = 1,etc.. (Ref. 2, "The odd perfect numbers", already

on our website. Perfect numbers, which are just the same, and the form 6k - 2, except

for the initial 6. Our demonstration Ref. 2).

In detail, the table for the first 10 perfect numbers with pairs S and P:

TAB. 3

PERFECT NUMBER S P

6 6 6

28 1 7

496 1 0

8128 1 2

33 550 336 1 0

8 589 869 056 1 0

137 438 691 328 1 0

2 305 843 008 139 952 128 1 0

2 658 455 991 569 831 744 654 692 615 953 842 176 1 0

191 561 942 608 236 107 294 793 378 084 303 638 130 997

321 548 169 2161 0

The sum S = 1 because all perfect numbers are derived from the Mersenne numbers:

The perfect number is given by 2p1

(2p1) with p prime number but all numbers of the

form 2m1

(2m1) withm odd integer give the sum to S = 1.

In fact, for the only perfect number 6 does not work because p = 2 and it is not an odd

integer.


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Moreover, all the perfect numbers end with digit 6 or 8.

By

2n

(2n+1

- 1)

We have:

2n

is even and ends with digit 2, 4, 8, 6

(2

n+1

- 1) is odd and ends with digit 3, 7, 5, 1.

The value '5 'should be discarded as the hypothesis would fall primality, then the pairs

that remain are (2,3), (4,7) and (6,1), whose products give the numbers 6 and 8, last

digit of each perfect number.


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2. CONCLUSIONS

There is NO probabilistic primality test secure and efficient way to determine whether a

number is prime or not.

The only certainty that we have is the following:

CONDITIONS NECESSARY BUT NOT SUFFICIENT to determine if a number N

is prime is that his last digit is equal to 1, 3, 7 or 9 and that the sum of its digits S is

not equal to 0, 3 or 6 because otherwise the number N is divisible by 3.

Equivalently we have that the only universal regularity that emerges is that all prime

numbers except 2 and 3 initials, are of the form 6k - 1 and 6k +1, since all otherpossible forms (6k-4, 6k-3. 6k-2, 6k) formed from composite numbers.

There are no other conditions and assumptions.

The sequence of prime numbers is intrinsically linked to increase to infinity of natural

numbers N+ and is completely devoid of any rule, it is precisely this "no rule" thatdistributes random prime numbers.

Even the famous Riemann hypothesis, surmising that it is true (it is yet to be proven)

does not lead to any connection of prime numbers with the placement of the zeros of the

Riemann zeta function.

Although the Real part of the zeros always worth , the Imaginary parts are of no help

to determine whether a number is prime or not.


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In addition, the zeta function, useless as a primality test as discussed above, it is useless

even to give aid to the factorization, for example that of the RSA numbers, as is oftenerroneously believed.

Some few algorithm assumes RH in truth there, but is not dangerous enough, in fact

encryption is RSA is still untouched. The zeta function does not give a list of prime

numbers, nor a list of pairs of prime numbers useful for this purpose. Lists of the pairs

give the twin primes conjecture, Goldbach's conjecture, Sophie Germain conjecture.

These conjectures are ongoing demonstration, with our contributions also present.

The products of two numbers twins of the same couple (or close to each other, ie withdifference 4, 6 etc..) are recognizable from the decimal part of their square root, veryhigh, eg. 0.999 ... and this type of product is easily factored by Fermat's algorithm used

in reverse, ie from the square root of N = p * q with p and q twins or very close. It is

located p immediately close to the square root of N (see our work on the assumption

percentage)


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3. ANALYSIS OF 1229 PRIMES LESS THAN 10000

In Table 4 the complete list of pairs S and P 1229 for all prime numbers that are in the

range 2 ... 10000

TAB. 4

N Prime Number S P1 2 2 2

2 3 3 3

3 5 5 5

4 7 7 7

5 11 2 1

6 13 4 3

7 17 8 7

8 19 1 0

9 23 5 610 29 2 0

11 31 4 3

12 37 1 3

13 41 5 4

14 43 7 3

15 47 2 1

16 53 8 6

17 59 5 0

18 61 7 619 67 4 6

20 71 8 7

21 73 1 3

22 79 7 0

23 83 2 6

24 89 8 0

25 97 7 0

26 101 2 0


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27 103 4 028 107 8 0

29 109 1 0

30 113 5 3

31 127 1 5

32 131 5 3

33 137 2 3

34 139 4 0

35 149 5 0

36 151 7 537 157 4 8

38 163 1 0

39 167 5 6

40 173 2 3

41 179 8 0

42 181 1 8

43 191 2 0

44 193 4 0

45 197 8 046 199 1 0

47 211 4 2

48 223 7 3

49 227 2 1

50 229 4 0

51 233 8 0

52 239 5 0

53 241 7 8

54 251 8 155 257 5 7

56 263 2 0

57 269 8 0

58 271 1 5

59 277 7 8

60 281 2 7

61 283 4 3

62 293 5 0


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99 523 1 3100 541 1 2

101 547 7 5

102 557 8 4

103 563 5 0

104 569 2 0

105 571 4 8

106 577 1 2

107 587 2 1

108 593 8 0109 599 5 0

110 601 7 0

111 607 4 0

112 613 1 0

113 617 5 6

114 619 7 0

115 631 1 0

116 641 2 6

117 643 4 0118 647 8 6

119 653 5 0

120 659 2 0

121 661 4 0

122 673 7 0

123 677 2 6

124 683 8 0

125 691 7 0

126 701 8 0127 709 7 0

128 719 8 0

129 727 7 8

130 733 4 0

131 739 1 0

132 743 5 3

133 751 4 8

134 757 1 2


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135 761 5 6136 769 4 0

137 773 8 3

138 787 4 5

139 797 5 0

140 809 8 0

141 811 1 8

142 821 2 7

143 823 4 3

144 827 8 4145 829 1 0

146 839 2 0

147 853 7 3

148 857 2 1

149 859 4 0

150 863 8 0

151 877 4 5

152 881 8 1

153 883 1 3154 887 5 7

155 907 7 0

156 911 2 0

157 919 1 0

158 929 2 0

159 937 1 0

160 941 5 0

161 947 2 0

162 953 8 0163 967 4 0

164 971 8 0

165 977 5 0

166 983 2 0

167 991 1 0

168 997 7 0

169 1009 1 0

170 1013 5 0


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171 1019 2 0172 1021 4 0

173 1031 5 0

174 1033 7 0

175 1039 4 0

176 1049 5 0

177 1051 7 0

178 1061 8 0

179 1063 1 0

180 1069 7 0181 1087 7 0

182 1091 2 0

183 1093 4 0

184 1097 8 0

185 1103 5 0

186 1109 2 0

187 1117 1 7

188 1123 7 6

189 1129 4 0190 1151 8 5

191 1153 1 6

192 1163 2 0

193 1171 1 7

194 1181 2 8

195 1187 8 2

196 1193 5 0

197 1201 4 0

198 1213 7 6199 1217 2 5

200 1223 8 3

201 1229 5 0

202 1231 7 6

203 1237 4 6

204 1249 7 0

205 1259 8 0

206 1277 8 8


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207 1279 1 0208 1283 5 3

209 1289 2 0

210 1291 4 0

211 1297 1 0

212 1301 5 0

213 1303 7 0

214 1307 2 0

215 1319 5 0

216 1321 7 6217 1327 4 6

218 1361 2 0

219 1367 8 0

220 1373 5 0

221 1381 4 6

222 1399 4 0

223 1409 5 0

224 1423 1 6

225 1427 5 2226 1429 7 0

227 1433 2 0

228 1439 8 0

229 1447 7 4

230 1451 2 2

231 1453 4 6

232 1459 1 0

233 1471 4 1

234 1481 5 5235 1483 7 6

236 1487 2 8

237 1489 4 0

238 1493 8 0

239 1499 5 0

240 1511 8 5

241 1523 2 3

242 1531 1 6


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243 1543 4 6244 1549 1 0

245 1553 5 3

246 1559 2 0

247 1567 1 3

248 1571 5 8

249 1579 4 0

250 1583 8 3

251 1597 4 0

252 1601 8 0253 1607 5 0

254 1609 7 0

255 1613 2 0

256 1619 8 0

257 1621 1 3

258 1627 7 3

259 1637 8 0

260 1657 1 3

261 1663 7 0262 1667 2 0

263 1669 4 0

264 1693 1 0

265 1697 5 0

266 1699 7 0

267 1709 8 0

268 1721 2 5

269 1723 4 6

270 1733 5 0271 1741 4 1

272 1747 1 7

273 1753 7 6

274 1759 4 0

275 1777 4 1

276 1783 1 6

277 1787 5 5

278 1789 7 0


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279 1801 1 0280 1811 2 8

281 1823 5 3

282 1831 4 6

283 1847 2 8

284 1861 7 3

285 1867 4 3

286 1871 8 2

287 1873 1 6

288 1877 5 5289 1879 7 0

290 1889 8 0

291 1901 2 0

292 1907 8 0

293 1913 5 0

294 1931 5 0

295 1933 7 0

296 1949 5 0

297 1951 7 0298 1973 2 0

299 1979 8 0

300 1987 7 0

301 1993 4 0

302 1997 8 0

303 1999 1 0

304 2003 5 0

305 2011 4 0

306 2017 1 0307 2027 2 0

308 2029 4 0

309 2039 5 0

310 2053 1 0

311 2063 2 0

312 2069 8 0

313 2081 2 0

314 2083 4 0


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315 2087 8 0316 2089 1 0

317 2099 2 0

318 2111 5 2

319 2113 7 6

320 2129 5 0

321 2131 7 6

322 2137 4 6

323 2141 8 8

324 2143 1 6325 2153 2 3

326 2161 1 3

327 2179 1 0

328 2203 7 0

329 2207 2 0

330 2213 8 3

331 2221 7 8

332 2237 5 3

333 2239 7 0334 2243 2 3

335 2251 1 2

336 2267 8 6

337 2269 1 0

338 2273 5 3

339 2281 4 5

340 2287 1 8

341 2293 7 0

342 2297 2 0343 2309 5 0

344 2311 7 6

345 2333 2 0

346 2339 8 0

347 2341 1 6

348 2347 7 6

349 2351 2 3

350 2357 8 3


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351 2371 4 6352 2377 1 6

353 2381 5 3

354 2383 7 0

355 2389 4 0

356 2393 8 0

357 2399 5 0

358 2411 8 8

359 2417 5 2

360 2423 2 3361 2437 7 6

362 2441 2 5

363 2447 8 8

364 2459 2 0

365 2467 1 3

366 2473 7 6

367 2477 2 5

368 2503 1 0

369 2521 1 2370 2531 2 3

371 2539 1 0

372 2543 5 3

373 2549 2 0

374 2551 4 5

375 2557 1 8

376 2579 5 0

377 2591 8 0

378 2593 1 0379 2609 8 0

380 2617 7 3

381 2621 2 6

382 2633 5 0

383 2647 1 3

384 2657 2 6

385 2659 4 0

386 2663 8 0


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387 2671 7 3388 2677 4 3

389 2683 1 0

390 2687 5 6

391 2689 7 0

392 2693 2 0

393 2699 8 0

394 2707 7 0

395 2711 2 5

396 2713 4 6397 2719 1 0

398 2729 2 0

399 2731 4 6

400 2741 5 2

401 2749 4 0

402 2753 8 3

403 2767 4 3

404 2777 5 2

405 2789 8 0406 2791 1 0

407 2797 7 0

408 2801 2 0

409 2803 4 0

410 2819 2 0

411 2833 7 0

412 2837 2 3

413 2843 8 3

414 2851 7 8415 2857 4 2

416 2861 8 6

417 2879 8 0

418 2887 7 5

419 2897 8 0

420 2903 5 0

421 2909 2 0

422 2917 1 0


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423 2927 2 0424 2939 5 0

425 2953 1 0

426 2957 5 0

427 2963 2 0

428 2969 8 0

429 2971 1 0

430 2999 2 0

431 3001 4 0

432 3011 5 0433 3019 4 0

434 3023 8 0

435 3037 4 0

436 3041 8 0

437 3049 7 0

438 3061 1 0

439 3067 7 0

440 3079 1 0

441 3083 5 0442 3089 2 0

443 3109 4 0

444 3119 5 0

445 3121 7 6

446 3137 5 0

447 3163 4 0

448 3167 8 0

449 3169 1 0

450 3181 4 6451 3187 1 6

452 3191 5 0

453 3203 8 0

454 3209 5 0

455 3217 4 6

456 3221 8 3

457 3229 7 0

458 3251 2 3


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459 3253 4 0460 3257 8 3

461 3259 1 0

462 3271 4 6

463 3299 5 0

464 3301 7 0

465 3307 4 0

466 3313 1 0

467 3319 7 0

468 3323 2 0469 3329 8 0

470 3331 1 0

471 3343 4 0

472 3347 8 0

473 3359 2 0

474 3361 4 0

475 3371 5 0

476 3373 7 0

477 3389 5 0478 3391 7 0

479 3407 5 0

480 3413 2 0

481 3433 4 0

482 3449 2 0

483 3457 1 6

484 3461 5 0

485 3463 7 0

486 3467 2 0487 3469 4 0

488 3491 8 0

489 3499 7 0

490 3511 1 6

491 3517 7 6

492 3527 8 3

493 3529 1 0

494 3533 5 0


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495 3539 2 0496 3541 4 6

497 3547 1 6

498 3557 2 3

499 3559 4 0

500 3571 7 6

501 3581 8 3

502 3583 1 0

503 3593 2 0

504 3607 7 0505 3613 4 0

506 3617 8 0

507 3623 5 0

508 3631 4 0

509 3637 1 0

510 3643 7 0

511 3659 5 0

512 3671 8 0

513 3673 1 0514 3677 5 0

515 3691 1 0

516 3697 7 0

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567 4127 5 2568 4129 7 0

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639 4733 8 0640 4751 8 5

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675 5039 8 0676 5051 2 0

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711 5393 2 0712 5399 8 0

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783 5987 2 0784 6007 4 0

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855 6619 4 0856 6637 4 0

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891 6947 8 0892 6949 1 0

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963 7583 5 3964 7589 2 0

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999 7907 5 01000 7919 8 0

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1035 8243 8 31036 8263 1 0

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1071 8599 4 01072 8609 5 0

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1100 8831 2 3

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1106 8867 2 6


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1107 8887 4 21108 8893 1 0

1109 8923 4 0

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1143 9221 5 01144 9227 2 0

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1161 9391 4 01162 9397 1 0

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1177 9497 2 0

1178 9511 7 0


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1179 9521 8 01180 9533 2 0

1181 9539 8 0

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1197 9697 4 01198 9719 8 0

1199 9721 1 0

1200 9733 4 0

1201 9739 1 0

1202 9743 5 0

1203 9749 2 0

1204 9767 2 0

1205 9769 4 0

1206 9781 7 01207 9787 4 0

1208 9791 8 0

1209 9803 2 0

1210 9811 1 0

1211 9817 7 0

1212 9829 1 0

1213 9833 5 0

1214 9839 2 0


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OBSERVATIONS

We note the frequent series of consecutive primes with P = 0

Examples: (in red those close to a power of 10, except 10^1)

From 89 a 109, with 29-23 = 6 consecutive prime numbers containing the digit 9

From 349 to 419, with 81-69 = 12 consecutive numbers, starting with the digits 33 and3*3 =9, with P = 0

From 907 a 1109, with 186 -154 = 32 consecutive primes, starting or containing the

digit 9

From 1879 to 2099, with 317-288 = 29 prime numbers containing a digit 9

Another group of consecutive prime numbers with P = 0 is located between 2897 and

3121, with 444-418 = 26 prime numbers, which also include the figure 9.

The most numerous series are close to a power of 10 (ie, a bit before and a little 'later,like 6 prime numbers around 100).

32 around in 1000 and 122 before 10,000 since 1129-1107 = 122 consecutive prime

numbers, starting of course with 8, but contain the digit 9, or starting with 9. An 'other

small series after 10 000 will complete the series for the fourth power of 10 = 10 ^ 4.

Any number containing the digit 9, with regard to P = 0, multiplied by all the other

digits of the number to be always a multiple of 9, so being modulo 9 = 0, its product isalways P = 0.

The same is true also for the numbers, prime or not prime, containing two-digit 3, also

not consecutive, because 3 * 3 = 9, and then all the products with two 3 digits in the

factors are congruent modulo 9 and then P = 0, and so also the digits 3 and 6, since 6 * 3

= 18 multiple of 9, and then the even its products with other digits of the number, and

hence P = 0 in all these cases.


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Let us now see a table with S and P related to prime numbers and composite of form 6k

-1 and 6k +1

Table 6

6k -1 primes andcompositeS P

6k+1 primes andcomposite

Observations for the pairs(S, P) for compositenumbers

55 5

71 7

112 1

134 3

178 7

191 0

235 6

257 1 7 1

29

11 2

31

4 3358 6

371 3 8 6

415 4

437 3

472 1

494 0 4 0

538 6

551 7 1 7

595 0

617 6

652 3

67

718 7

731 3

775 0

797 0 5 0


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Last column:

7 1

8 6

4 0

1 7

5 0

1 0

5 0

7 5

2 0

8 1


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7 1 exists only once8 6 exists only once4 0 exists only once1 7 exists only once5 0 exists only once1 0 exists only once7 5 exists only once

2 0 exists only once8 1 exists only once

Conclusions:

If a number N of such forms 6k-1 and 6k + 1 has one of these pairs, it is probably

composite. For example, we see the multiple of 7, then 7 * p

Table 7

7*p Pairs S and P Observations

35 8 6

49 4 0

77 5 0

91 1 0

119 2 0

161 8 6

203 5 0

217 1 5


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Pairs for numbers N = 7*p

8 6

4 0

5 0

1 0

2 0

8 6

repetition4 0repetition

List of pairs for composite numbers (from Table 6)

7 1 exists only once8 6 exists only once

4 0 exists only once1 7 exists only once5 0 there is twice1 0 exists only once7 5 exists only once2 0 exists only once8 1 exists only once

The two lists match, although in a different order, in the second list is missing 1 5,which appears for the first time for

N = 7*31 =217

Same thing could be done to the multiple of 11, up to p = 31


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11*p Pairs S and P Observations

55 1 7

77 5 0

121 4 2

143 8 3

187 7 2

209 2 0

253 1 3

341 8 3

Some couples (in brown) are repeated, others appear new (in purple)

Pairs (S, P)

1 7

5 0

4 2

8 37 2

2 0

1 3

8 3

The pairs (1, 7), (5, 0) and (2, 0) repeat for 11 * p, so there may be repeated for some of

the next multiple, and therefore are not evidence of their primality.

For example, extending the table to 11 * p, we have


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11*p Pairs (S, P) Observations

55 1 7

77 5 0

121 4 2

143 8 3

187 7 2

209 2 0

253 1 3

341 (= 11*31) 8 3repetition

Seconda parte

407 (=11*39) 2 0repetition

451 1 2

473 5 3

517 4 8

561 3 0

737 8 3repetition

781 7 6

803 (= 11*73) 2 0repetition

869 (=11*79) 5 0repetition

Pairs first part

Pairs (S, P

1 7

5 0

4 2

8 3

7 2

2 0

1 3


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Pairs second part

2 0repetition

1 2

5 3

4 8

3 08 3repetition

7 6

2 0repetition

5 0repetition

(5. 0), (8, 3), (2, 0), are the pairs (S, P) most frequent for numbers N multiple of 11. Forexample, if we have the number 1067 to be tested, we find the pair (5, 0), the second of

this list, and then 1067 may be not a prime number, and a multiple of 11, in fact 1067 =11 * 97. Additional tables for multiple other primes might allow to identify their pairs

(S, P) "favorite", and testing them, we would find that often are not prime numbers, but

particular multiple of a prime number (11 in this case) Another example:

N = 1397,

a pair (2, 0), present a favorite of 11 * p

In fact, 1397 = 11 * 127.

These are useful, but not deterministic primality test or not primality test.


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CONCLUSIONS

On this work we have discovered something about the primality test, which may be

explored in the future with both our or others further research.


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4. REFERENCES

1a) Test di primalit AKS, da Wikipedia :

Algoritmo AKS

Da Wikipedia, l'enciclopedia libera.

Questa voce sull'argomento teoria dei numeri solo un abbozzo.

Contribuisci a migliorarla secondo le convenzioni di Wikipedia.

L'algoritmo AKS (dalle iniziali dei tre ideatori, i matematici indiani Manindra Agrawal,Neeraj Kayal e Nitin Saxena) un test di primalit di complessit polinomiale. In

particolare, l'algoritmo ha tempo di esecuzione O(log12+

n), mentre una variante

proposta nel 2005 da Carl Pomerance e Hendrik Lenstra ha complessit O(log6+

n)

Pubblicato nel 2002, ha fruttato ai suoi scopritori diversi premi, tra cui il premio Gdel

e il premio Fulkerson nel 2006.

1b) AKS primality test (in inglese)From Wikipedia, the free encyclopedia

Jump to: navigation, search

The AKS primality test (also known as AgrawalKayalSaxena primality test and

cyclotomic AKS test) is a deterministic primality-proving algorithm created and

published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at

the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled

"PRIMES is in P".[1]

The algorithm determines whether a number is prime or composite

within polynomial time. The authors received the 2006 Gdel Prize and the 2006

Fulkerson Prize for this work.Contents

1 Importance

2 Concepts

3 History and running time4 Algorithm

5 References

6 Further reading

7 External links


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ImportanceAKS is the first primality-proving algorithm to be simultaneously general, polynomial,

deterministic, and unconditional. Previous algorithms had been developed for centuriesbut achieved three of these properties at most, but not all four.

The AKS algorithm can be used to verify the primality of any general number given.

Many fast primality tests are known that work only for numbers with certain properties.

For example, the LucasLehmer test for Mersenne numbers works only for Mersenne

numbers, while Ppin's test can be applied to Fermat numbers only.

The maximum running time of the algorithm can be expressed as a polynomial over the

number of digits in the target number. ECPP and APR conclusively prove or disprovethat a given number is prime, but are not known to have polynomial time bounds for allinputs.

The algorithm is guaranteed to distinguish deterministically whether the target numberis prime or composite. Randomized tests, such as MillerRabin and BailliePSW, can

test any given number for primality in polynomial time, but are known to produce only

a probabilistic result.

The correctness of AKS is not conditional on any subsidiary unproven hypothesis. In

contrast, the Miller test is fully deterministic and runs in polynomial time over all

inputs, but its correctness depends on the truth of the yet-unproven generalized Riemannhypothesis.

Concepts

The AKS primality test is based upon the following theorem: An integer n ( 2) is

prime if and only if the polynomial congruence relation

holds for all integers a coprime to n (or even just for some such integer a, in particular

for a = 1).[1]

Note that x is a free variable. It is never substituted by a number; instead

you have to expand and compare the coefficients of the x powers.

This theorem is a generalization to polynomials of Fermat's little theorem, and caneasily be proven using the binomial theorem together with the following property of the

binomial coefficient:

for all if and only if n is prime.

While the relation (1) constitutes a primality test in itself, verifying it takes exponential

time. Therefore, to reduce the computational complexity, AKS makes use of the related

congruence


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which is the same as:

for some polynomials f and g. This congruence can be checked in polynomial time withrespect to the number of digits in n, because it is provable that r need only be

logarithmic with respect to n. Note that all primes satisfy this relation (choosing g = 0 in

(3) gives (1), which holds for n prime). However, some composite numbers also satisfy

the relation. The proof of correctness for AKS consists of showing that there exists a

suitably small r and suitably small set of integers A such that, if the congruence holds

for all such a in A, then n must be prime.

2) I NUMERI PERFETTI DISPARI

(proposta di dimostrazione della loro inesistenza)

Gruppo B. Riemann

Michele Nardelli, Francesco Di Noto

Aggiungiamo un generatore di numeri primi creato da

(Marco Cavicchioli) e pubblicato sul sito

(http://www.MarcoCavicchioli.it/numeri_primi_generatore.html

3) Generatore di numeri primi in Javascript

di Marco Cavicchioli

Inserire un numero intero positivo (N) maggiore di 1 nel seguente campo e cliccare su

"Genera":Inizio modulo

N =79

Fine modulo

Una volta cliccato su "Genera" un codice Javascript estrarr dalla successione dei

numeri interi positivi minori o uguali a N l'elenco dei numeri primi in essa riconosciuti.

Dato che la funzione Javascript che esegue ci genera (n-1) successioni da (n-1) a (n-1)2

la sua esecuzione risulta molto lenta per N molto alti (per esempio oltre 1000).

Da questo elenco convenzionalmente escluso il numero 1, perch tutti i numeri interi

positivi sono multipli di 1.Per vedere il codice Javascript sufficiente aprire questa pagina html con un editor di


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testo. Per tenere questo codice il pi pulito possibile ho scelto di non includere in questapagina alcun elemento grafico.

Ovviamente tutto il materiale contenuto in questa pagina (sia testo che codice) CopyLeft, ovvero disponibile liberamente a chiunque. L'autore tuttavia richiede

espressamente che, qualora venisse utilizzato, citato o ripubblicato, in ogni luogo e

forma, vengano sempre riportati con chiarezza ed evidenza il nome dell'autore (Marco

Cavicchioli) e l'url (http://www.MarcoCavicchioli.it/numeri_primi_generatore.html).

Questa pagina ottimizzata per Mozilla Firefox.

UPDATE: segnalo che stato pubblicato un TEOREMA FONDAMENTALE DEI

NUMERI PRIMI che merita di essere letto.

Descrizione della funzione

I numeri interi positivi maggiori di 1 sarebbero in realt tutti numeri multipli (di 1

stesso, che il primo numero intero positivo, e quello da cui vengono generati per

somma tutti gli altri numeri interi positivi). Per "numero primo" invece si intende un

numero che non multiplo (ovvero: un numero intero positivo o multiplo, o primo). Faccio notare che, a rigor di logica, a questo punto 1 sarebbe l'unico numeroprimo, perch l'unico a non essere multiplo (dato che tutti gli altri numeri interi positivi

sono perlomeno multipli di 1). Ma questo non esclude che si possa procedere lo stesso,con un'apposita accortezza...

Estrarre la successione dei numeri primi minori o uguali a N pertanto significa estrarre

dalla successione di tutti i numeri interi positivi minori o uguali a N quelli che non siano

soltanto multipli di 1. Questo concetto presume che si escluda la possibilit che un

numero N possa essere multiplo di s stesso. Per tentare di convincervi di ci devo fare

un passo indietro.

Se accettiamo il presupposto che tutti i numeri interi positivi siano in realt funzioni di 1

(ovvero valori risultanti dall'applicazione di una data funzione somma al numero 1,come se in realt esistessero solo due numeri "primordiali", 0 e 1, con tutti i restanti

numeri interi superiori in realt ricavati da somme di 1, e pertanto "multipli di 1") allora

tutti i numeri interi positivi, con l'esclusione di 1, sarebbero somme di "uni" (es. 2=1+1,

3=1+1+1, 7=1+1+1+1+1+1+1, ecc.). A questo punto potremmo definire "multipli"

(sempre con l'esclusione di 1 stesso) tutti i numeri interi positivi che si ottengono

sommando ripetutamente dei "pattern di uni". Per pattern di uni intendo sequenze

identiche ripetute di n uni (ad esempio (1+1+1)+(1+1+1)+(1+1+1)=9 dimostra che 9

un numero multiplo). Se imponiamo che, in questo caso, n sia superiore a 1 allora

otteniamo che i numeri primi sono quei numeri che non si ottengono da sommeripetute di pattern di n uni, con n>1. Il fatto che sia richiesto che tali somme sano


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ripetute esclude che un numero intero positivo N possa essere multiplo di s stesso,perch in questo caso il pattern di uni non potrebbe essere ripetutamente sommato per

ottenere N, se n vale N.Pertanto posso ora elencare alcune considerazioni importanti:

i numeri multipli sono quei numeri interi positivi che si ottengono da somme ripetute di

pattern di n uni, con n>1

dato che tali somme di pattern devono essere ripetute, un numero intero positivo N non

pu essere multiplo di s stesso, perch n deve essere minore di N (n


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4) REGOLA UNIVERSALE PER TROVARE TUTTI I NUMERI

PRIMI

Ing. Pier Francesco Roggero, Dott. Michele Nardelli, Francesco Di Noto

gi sul nostro sito http://nardelli.xoom.it/virgiliowizard/

pier francesco roggero, michele nardelli, francesco di noto - "there is no secure test for...

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