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- 11:50, 4 May 2023 Richard47 talk contribs created page User:Richard47/Fake Proof for 1=2 (Created page with "Prove that 1 = 2: First, we prove that the series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... is absolutely convergent: Let A = sigma(1/n), n runs the integers not containing "9453" as substring then A = (1 + 1/2 + 1/3 + 1/4 + ... + 1/9451 + 1/9452 + 1/9454 + 1/9455 + 1/9456 + ... + 1/10000) + (1/10001 + 1/10002 + 1/10003 + ... + 1/19451 + 1/19452 + 1/19454 + 1/19455 + 1/19456 + ... + 1/29451 + 1/29452 + 1/29454 + 1/29455 + 1/29456 + ... + 1/94528 + 1/94529 + 1/94540 + 1/9454...")
- 03:08, 16 April 2023 Richard47 talk contribs created page Numbers with All Digits Have a Common Factor are Divisible by This Factor (We really need trivial proofs, just like the proof Null String has Length Zero)
- 00:04, 14 March 2023 Richard47 talk contribs created page Category:Near-repdigit Primes (Created page with "Category:Prime Numbers")
- 00:03, 14 March 2023 Richard47 talk contribs created page Category:Near-repdigit Primes/Examples (Created page with "Category:Near-repdigit Primes")
- 00:02, 14 March 2023 Richard47 talk contribs created page Category:555,555,555,551 (Created page with "Category:Specific Numbers")
- 23:59, 13 March 2023 Richard47 talk contribs created page 555,555,555,551 (Created page with "{{NumberPageLink|prev = 555,555,555,550|next = 555,555,555,552}} == Number == $555 \, 555 \, 555 \, 551$ ('''sixty-six million, six hundred thousand and forty-nine''') is: :The $21, \, 366, \, 409, \, 911$th prime number :The 412nd near-repdigit prime :The $76$th minimal prime base $10$ after $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $4...")
- 23:54, 13 March 2023 Richard47 talk contribs created page Definition:Proper String (Redirected page to Definition:Substring/Proper) Tag: New redirect
- 23:53, 13 March 2023 Richard47 talk contribs created page Definition:Substring/Proper (Created page with "== Definition == Let $\LL$ be a formal language with alphabet $\AA$. Let $S$ be a string in $\AA$. Let $T$ be a string in $\AA$ such that: :$S = S_1 T S_2$ where: :$S_1$ and $S_2$ are strings in $\AA$ (possibly null); :$S_1 T S_2$ is the concatenation of $S_1...")
- 23:46, 13 March 2023 Richard47 talk contribs created page Definition:Near-repdigit Prime (Created page with "Near-repdigit prime are Definition:Prime Number primes with $\ge 3$ digits with only one digit different from others, the first few such primes are 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997, 1117, 1151, 1171, 1181, 1511, ... {{OEIS|A164937}}")
- 23:41, 13 March 2023 Richard47 talk contribs created page Definition:Wall-Sun-Sun Prime (Created page with "== Definition == <onlyinclude> A '''Wall-Sun-Sun prime''' is a prime number $p$ such that: :$p^2 \divides F_{p-(p|5)}$ where $\divides$ denotes divisibility. </onlyinclude>")
- 23:39, 13 March 2023 Richard47 talk contribs created page Definition:Rhonda Number (Created page with "A positive integer n is called a base-*b* Rhonda number if the product of the base-*b* digits of *n* is equal to *b* times the sum of *n*'s prime factors. These numbers were named by K. S. Brown after an acquaintance of his whose residence number *25662* satisfies this property. The etymology of the term is therefore similar to the Smith numbers.")
- 23:38, 13 March 2023 Richard47 talk contribs created page Definition:Sum-Product Number (Created page with "A sum-product number in a given number base *b*, is a natural number that is equal to the product of the sum of its digits and the product of its digits.")
- 23:36, 13 March 2023 Richard47 talk contribs created page Definition:Deletable Prime (Created page with "A deletable prime is a prime number which has the property that deleting digits one at a time in some order gives a prime at each step. For example, 410256793 is a deletable prime since each member of the sequence 410256793, 41256793, 4125673, 415673, 45673, 4567, 467, 67, 7 is prime.")
- 23:35, 13 March 2023 Richard47 talk contribs created page Definition:Weakly Prime (Created page with "A prime number is said to be weakly prime if changing a single digit to every other possible digit produces a composite number when performed on each digit. The first few such numbers are 294001, 505447, 584141, 604171, 971767, 1062599, ...")
- 23:35, 13 March 2023 Richard47 talk contribs created page Category:5,000,000,000,000,000,000,000,000,000,027 (Created page with "Category:Specific Numbers")
- 23:34, 13 March 2023 Richard47 talk contribs created page 5,000,000,000,000,000,000,000,000,000,027 (Created page with "{{NumberPageLink|prev = 5,000,000,000,000,000,000,000,000,000,026|next = 5,000,000,000,000,000,000,000,000,000,028}} == Number == $5,000,000,000,000,000,000,000,000,000,027$ ('''five nonillion and twenty-seven''') is: :The $77$th (and last) minimal prime base $ after $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$, $67$, $71$, $73$, $79$, $83$, $89$, $97$, $227$, $251$,...")
- 23:04, 13 March 2023 Richard47 talk contribs created page Definition:Empty String (Created page with "== Definition == <onlyinclude> The '''empty string''' is a string which has no characters.")
- 23:03, 13 March 2023 Richard47 talk contribs created page Numbers with All Digits Have a Common Factor are Divisible by This Factor (Created page with "== Theorem == <onlyinclude> A number expressed in decimal notation is divisible by $d$ if all its digits are divisible by $d$. That is: :$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $d$ if :$gcd(a_0,a_1,\ldots,a_n)$ is Definition:Divisor of Integer|divisi...")
- 22:27, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of 47 is of Maximal Length (Created page with "== Theorem == The decimal expansion of the reciprocal of $47$ has the maximum period, that is: $46$: :$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$ {{OEIS|A021051}} Category:47 Category:Examples of Reciprocals")
- 21:57, 11 March 2023 Richard47 talk contribs created page Talk:Minimal Prime/Various Number Bases (Created page with "This page is I researched the minimal primes in other bases, and I added the condition that the primes must be greater than the base to make the problem more interesting, e.g. I want to including "finding the smallest prime of the form k*b^n+1" (or proving that such prime does not exist) and "finding the smallest prime of the form k*b^n-1" (or proving that such prime does not exist) for all natural number k < b. --~~~~")
- 17:16, 11 March 2023 Richard47 talk contribs created page There are 77 Minimal Primes in Base 10 if Single-Digit-Primes Subsequences are Allowed (Created page with "== Theorem == There are $77$ minimal primes in base $: :$11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$, $67$, $71$, $73$, $79$, $83$, $89$, $97$, $227$, $251$, $257$, $277$, $281$, $349$, $409$, $449$, $499$, $521$, $557$, $577$, $587$, $727$, $757$, $787$, $821$, $827$, $857$, $877$, $881$, $887$, $991$, $2087$, $2221$, $5051$, $5081$, $5501$, $5581$, $5801$, $5851$, $6...")
- 13:36, 11 March 2023 Richard47 talk contribs created page Reciprocal of 9801 (Created page with "== Theorem == The reciprocal of $9801$ is: <onlyinclude> :$\dfrac 1 {9801} = 0 \cdotp \dot 00010 \, 20304 \, 05060 \, 70809 \, 10111 \, 21314 \, ... \, 94959 \, 6979 \dot 9$ </onlyinclude> {{OEIS|A034948}} Its period is the consecutive of 00, 01, 02, ..., 99 except 98. Category:9801 Category:Examples of Reciprocals")
- 13:17, 11 March 2023 Richard47 talk contribs created page Category:1111111111111111111 (Created blank page)
- 13:08, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of 73 has Length 8 (Created page with "== Theorem == $73$ is the first positive integer the decimal expansion of whose reciprocal has a period of $8$: :$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$ {{OEIS|A021077}} Category:73 Category:Examples of Reciprocals")
- 13:04, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of Repunit 19 is 19 (Why not consider R19?)
- 12:57, 11 March 2023 Richard47 talk contribs created page Reciprocal of 121 (Created page with "== Theorem == <onlyinclude> The decimal expansion of the reciprocal of $121$ has a particularly interesting pattern: :$\dfrac 1 {121} = 0 \cdotp \dot 008 \, 264 \, 462 \, 809 \, 917 \, 355 \, 371 \, \dot 9$ </onlyinclude> {{OEIS|A021125}} == Also see == * Reciprocal of Square of 1 More than Number Base Category:121 Category:Examples of Reciprocals")
- 12:53, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of 83 is of Prime Length (Created page with "== Theorem == The decimal expansion of the reciprocal of $83$ has an prime period, that is: $41$: <onlyinclude> :$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$ {{OEIS|A021087}} </onlyinclude> It is the smallest prime number to have an Definition:Prime Numbe...")
- 12:44, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of 101 has Length 4 (Created page with "== Theorem == $101$ is the first positive integer the decimal expansion of whose reciprocal has a period of $4$: :$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$ {{OEIS|A021105}} Category:Examples of Reciprocals")
- 12:43, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of 79 is of Sixth Maximal Length (Created page with "== Theorem == The decimal expansion of the reciprocal of $79$ has $\dfrac 1 6$ the maximum period, that is: $13$: :$\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$ {{OEIS|A021083}} Category:79 Category:Examples of Reciprocals")
- 12:38, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of 29 is of Maximal Length (Created page with "== Theorem == The decimal expansion of the reciprocal of $29$ has the maximum period, that is: $28$: :$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ {{OEIS|A021033}} Category:29 Category:Examples of Reciprocals")
- 12:36, 11 March 2023 Richard47 talk contribs created page Period of Reciprocal of 43 is of Odd Length (Created page with "== Theorem == The decimal expansion of the reciprocal of $43$ has an odd period, that is: $21$: <onlyinclude> :$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$ {{OEIS|A021047}} </onlyinclude> Category:43 Category:Examples of Reciprocals")
- 12:32, 11 March 2023 Richard47 talk contribs created page Reciprocal of 3 (Created page with "== Theorem == The decimal expansion of the reciprocal of $3$ has a period of $1$: :$\dfrac 1 {3} = 0 \cdotp \dot 3$ {{OEIS|A010701}} Category:3 Category:Examples of Reciprocals")
- 12:30, 11 March 2023 Richard47 talk contribs created page Reciprocal of 11 (Created page with "== Theorem == The decimal expansion of the reciprocal of $11$ has a period of $2$: :$\dfrac 1 {11} = 0 \cdotp \dot 0 \dot 9$ {{OEIS|A010680}} Category:11 Category:Examples of Reciprocals")
- 12:28, 11 March 2023 Richard47 talk contribs created page Reciprocal of 5 (Created page with "== Theorem == The reciprocal of the prime number 5 has no period, it is 0.2 Category:5 Category:Examples of Reciprocals")
- 12:27, 11 March 2023 Richard47 talk contribs created page Reciprocal of 2 (Created page with "== Theorem == The reciprocal of the prime number 2 has no period, it is 0.5 Category:2 Category:Examples of Reciprocals")
- 22:38, 10 March 2023 Richard47 talk contribs created page Aurifeuillian Factorization/Examples/3^6n+3 + 1 (Created page with "== Example of Aurifeuillian Factorization == <onlyinclude> :$3^{6 n + 3} + 1 = \paren {3^{2 n + 1}} \paren {3^{2 n + 1} - 3^{n + 1} + 1} \paren {3^{2 n + 1} + 3^{n + 1} + 1}$ </onlyinclude>")
- 21:55, 10 March 2023 Richard47 talk contribs created page Left-Truncatable Prime/Various Number Bases (Created page with "== Right-Truncatable Prime: Various Number Bases == {{tidy}} <onlyinclude> All numbers are written in base ''b'', using A−Z to represent digit values 10 to 35. Base 2: None (but 1, 11, 111 are left-truncatable primes if you count 1 as prime) Base 3: {2, 12, 212} (3 left-truncatable primes) Base 4: {2, 3, 13, 23, 113, 223, 323, 1223, 2113, 3323, 21223, 32113, 33323, 233323, 321223, 333323} (16 left-truncatable primes) Base 5:...")
- 21:44, 10 March 2023 Richard47 talk contribs created page Long Period Prime/Examples (Created page with "7 is a long period prime, since 1/7 = $0 \cdotp \dot 14285 \, \dot 7$ has period 6 = 7-1 17 is a long period prime, since 1/17 = $0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ has period 16 = 17-1 19 is a long period prime, since 1/19 = $0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ has period 18 = 19-1 23 is a long period prime, since 1/23 = $0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ has period 22 = 23-1 29 is a long period prime, since 1/...")
- 21:28, 10 March 2023 Richard47 talk contribs created page Period of Reciprocal of 41 has Length 5 (Created page with "== Theorem == $41$ is the first positive integer the decimal expansion of whose reciprocal has a period of $5$: :$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$ {{OEIS|A021045}} Category:Examples of Reciprocals")
- 11:57, 10 March 2023 Richard47 talk contribs created page Definition:Hexatrigesimal Notation (Created page with "== Definition == <onlyinclude> '''Hexatrigesimal notation''' is the technique of expressing numbers in base $36$. Every number $x \in \R$ is expressed in the form: :$\ds x = \sum_{j \mathop \in \Z} r_j 36^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, \ldots, 35}$ </onlyinclude> In order to be able to represent numbers in such a format conveniently and readably, it is necess...")
- 11:53, 10 March 2023 Richard47 talk contribs created page Long Period Prime/Various Number Bases (Created page with "In base ''b'', the long period primes are the primes not dividing ''b'' which have ''b'' as a primitive root {|class=wikitable !base !long period primes |- |2 |3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, ... |- |3 |2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 1...")
- 11:44, 10 March 2023 Richard47 talk contribs created page Definition:Undecimal Notation (Created page with "== Definition == <onlyinclude> '''Undecimal notation''' is the technique of expressing numbers in base $. Every number $x \in \R$ is expressed in the form: :$\ds x = \sum_{j \mathop \in \Z} r_j 11^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, \ldots, 9, 10}$ </onlyinclude> In order to be able to represent numbers in such a format conveniently and readably, it is necessar...")
- 11:43, 10 March 2023 Richard47 talk contribs created page Definition:Nonary Notation (Created page with "== Definition == <onlyinclude> '''Septenary notation''' is the technique of expressing numbers in base $. That is, every number $x \in \R$ is expressed in the form: :$\ds x = \sum_{j \mathop \in \Z} r_j 9^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7, 8}$ </onlyinclude> Category:Definitions/Number Bases")
- 11:42, 10 March 2023 Richard47 talk contribs created page Definition:Septenary Notation (Created page with "== Definition == <onlyinclude> '''Septenary notation''' is the technique of expressing numbers in base $. That is, every number $x \in \R$ is expressed in the form: :$\ds x = \sum_{j \mathop \in \Z} r_j 7^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6}$ </onlyinclude> Category:Definitions/Number Bases")
- 11:41, 10 March 2023 Richard47 talk contribs created page Definition:Senary Notation (Created page with "== Definition == <onlyinclude> '''Senary notation''' is the technique of expressing numbers in base $. That is, every number $x \in \R$ is expressed in the form: :$\ds x = \sum_{j \mathop \in \Z} r_j 6^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5}$ </onlyinclude> Category:Definitions/Number Bases")
- 11:40, 10 March 2023 Richard47 talk contribs created page Definition:Quaternary Notation (Created page with "== Definition == <onlyinclude> '''Quaternary notation''' is the technique of expressing numbers in base $. That is, every number $x \in \R$ is expressed in the form: :$\ds x = \sum_{j \mathop \in \Z} r_j 4^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3}$ </onlyinclude> Category:Definitions/Number Bases")
- 20:09, 9 March 2023 Richard47 talk contribs created page All Numbers of the Form 28000...0007 are Divisible by 7 (Created page with "The value of 28000...0007 (with ''n'' 0's) is 2×10<sup>''n''+2</sup>+8×10<sup>''n''+1</sup>+0×10<sup>''n''</sup>+...+0×10<sup>1</sup>+7 = 28×10<sup>''n''+1</sup>+7 {{begin-eqn}} {{eqn | l = 28 \times 10^{n+1} + 7 | o = \equiv | r = 0 \times 3^{n+1} + 0 | rr = \pmod 7 | c = Fermat's Little Theorem and Congruence of Powers }} {{eqn | o = \equiv | r = 0 + 0 | rr = \pmod 7 | c = Congruence of Pr...")
- 13:57, 9 March 2023 Richard47 talk contribs created page There are 4260 Left-Truncatable Primes in Base 10 (Created page with "== Theorem == In base $, there are $4260$ right-truncatable primes: :2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, 1223, ..., 357686312646216567629137 == Proof == Of the $1$-Definition:Digi...")
- 11:32, 9 March 2023 Richard47 talk contribs created page All Numbers of the Form 4666...6669 are Divisible by 7 (Created page with "The value of 4666...6669 (with ''n'' 0's) is (14×10<sup>''n''+1</sup>+7)/3 {{begin-eqn}} {{eqn | l = (14 \times 10^{n+1} + 7) \div 3 | o = \equiv | r = (0 \times 3^{n+1} + 0) \div 3 | rr = \pmod 7 | c = Fermat's Little Theorem and Congruence of Powers }} {{eqn | o = \equiv | r = (0 + 0) \div 3 | rr = \pmod 7 | c = Congruence of Product }} {{eqn | o = \equiv | r = 0 \div 3 | rr = \...")
- 11:27, 9 March 2023 Richard47 talk contribs created page All Numbers of the Form 2000...0001 are Divisible by 3 (Refo There are 77 Minimal Primes in Base 10, we need to prove that some families contain no primes)