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-   -   Two prime-related conjectures (https://www.mersenneforum.org/showthread.php?t=27307)

 sweety439 2021-11-09 11:02

Two prime-related conjectures

[B]Conjecture 1 (conjecture about square numbers and odd primes)[/B]: Every number which is not twice a square number ([URL="https://oeis.org/A001105"]A001105[/URL]) can be written as (twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 47 known counterexamples, the largest known counterexample is 43358, and I conjectured that all other numbers which is not twice a square number can be written as this form.

[CODE]
1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358
[/CODE]

(the numbers in this set are:

* odd numbers: [URL="https://oeis.org/A060003"]A060003[/URL] (including: 1, odd primes [URL="https://oeis.org/A042978"]A042978[/URL] \ {2}, odd composites {5777, 5993} (Goldbach incorrectly conjectured that no such odd composites exist, see [URL="https://www.jstor.org/stable/2690477"]https://www.jstor.org/stable/2690477[/URL])) [10 elements]

* even numbers where n/2 is prime: 2*[URL="https://oeis.org/A065377"]A065377[/URL] U {6} (3 is the only number when written as sum of square number and prime requiring the even prime 2, this is equivalent to 6=2*3 in this problem) [16 elements]

* even numbers where n/2 is composite: 2*[URL="https://oeis.org/A020495"]A020495[/URL] [21 elements]

totally 10+16+21 = 47 elements, see the OEIS references, [URL="https://oeis.org/A060003"]odd numbers have been checked to 2*10^13[/URL], even numbers where n/2 is prime have been checked to 2*10^9 (since [URL="https://oeis.org/A065377"]"sum of square and prime" have been checked to 10^9[/URL]), even numbers where n/2 is composite have been checked to 2*10^11 (since [URL="https://oeis.org/A020495"]"sum of square and prime" have been checked to 10^11[/URL]), thus my conjecture about square numbers and odd primes have been checked to 2*10^9)

[B]Conjecture 2 (conjecture about triangular numbers and odd primes)[/B]: Every number which is not twice a triangular number ([URL="https://oeis.org/A002378"]A002378[/URL]) can be written as (twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 8 known counterexamples, the largest known counterexample is 432, and I conjectured that all other numbers which is not twice a triangular number can be written as this form.

[CODE]
1, 3, 4, 10, 14, 122, 422, 432
[/CODE]

(the numbers in this set are:

* odd numbers: {1, 3}, no OEIS sequences for this, and (conjectured by Zhi-Wei Sun, see [URL="https://oeis.org/A144590"]A144590[/URL]) every odd number larger than 3 can be written as sum of a nonzero Pronic number and an odd prime [2 elements]

* even numbers where n/2 is prime: 2*[URL="https://oeis.org/A065397"]A065397[/URL] U {10} (5 is the only number when written as sum of triangular number and prime requiring the even prime 2, this is equivalent to 10=2*5 in this problem) [5 elements]

* even numbers where n/2 is composite: {432} (Zhi-Wei Sun conjectured that 432 is the only such number, see [URL="https://arxiv.org/abs/0803.3737"]https://arxiv.org/abs/0803.3737[/URL]) [1 element]

totally 2+5+1 = 8 elements, see the OEIS references, [URL="https://oeis.org/A144590"]odd numbers have been checked to 10^10[/URL], even numbers where n/2 is prime have been checked to 8*10^9 (since [URL="https://oeis.org/A255904"]"sum of triangular and prime" have been checked to 4*10^9[/URL]), even numbers where n/2 is composite have been checked to 2*10^12 (since [URL="https://oeis.org/A132399"]"sum of triangular and prime" have been checked to 10^12[/URL]), thus my conjecture about triangular numbers and odd primes have been checked to 8*10^9)

Note:

"U": [URL="https://en.wikipedia.org/wiki/Union_(set_theory)"]set union[/URL]
"\": [URL="https://en.wikipedia.org/wiki/Complement_(set_theory)#Relative_complement"]set difference[/URL]

 sweety439 2021-11-09 11:22

There are many numbers which is not twice a square number which can be [B]uniquely[/B] written as (twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, but I conjectured there are only finite such numbers (there are 318 known such numbers, the largest such number is 89072), furthermore, for every number r>=0, there are only finitely many numbers which is not twice a square number which can be written as (twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, no more than r ways.

[CODE]
5, 7, 9, 11, 12, 14, 16, 22, 23, 27, 29, 30, 33, 34, 36, 38, 41, 44, 47, 48, 52, 53, 57, 58, 59, 65, 71, 80, 83, 86, 92, 95, 100, 102, 107, 110, 113, 123, 140, 143, 146, 148, 149, 152, 158, 161, 164, 188, 194, 197, 198, 212, 218, 230, 233, 239, 240, 257, 266, 272, 278, 281, 284, 287, 290, 302, 308, 314, 317, 318, 323, 332, 340, 347, 356, 362, 368, 383, 386, 404, 407, 410, 413, 422, 438, 442, 443, 446, 449, 458, 470, 482, 492, 500, 506, 530, 536, 542, 548, 554, 566, 569, 590, 596, 602, 620, 626, 632, 638, 650, 656, 662, 668, 680, 692, 698, 743, 773, 782, 785, 788, 794, 798, 818, 824, 848, 863, 872, 884, 890, 926, 938, 980, 998, 1010, 1022, 1082, 1094, 1118, 1124, 1148, 1172, 1178, 1220, 1227, 1232, 1238, 1292, 1322, 1367, 1388, 1415, 1418, 1478, 1502, 1562, 1586, 1598, 1622, 1668, 1670, 1703, 1748, 1754, 1787, 1828, 1844, 1892, 1898, 1940, 1958, 1988, 2042, 2060, 2090, 2123, 2132, 2138, 2174, 2180, 2210, 2234, 2328, 2342, 2402, 2408, 2438, 2486, 2507, 2558, 2582, 2648, 2708, 2732, 2762, 2768, 2822, 2858, 2900, 2933, 3002, 3062, 3110, 3242, 3284, 3317, 3434, 3452, 3482, 3515, 3530, 3572, 3620, 3662, 3713, 3722, 3758, 3770, 3962, 3980, 3998, 4022, 4082, 4118, 4148, 4178, 4292, 4334, 4490, 4502, 4532, 4538, 4568, 4586, 4673, 4688, 4820, 4832, 4958, 5078, 5102, 5300, 5612, 5642, 5708, 5798, 5852, 5942, 5987, 6008, 6188, 6218, 6302, 6332, 6368, 6518, 6602, 6797, 6836, 6938, 7004, 7142, 7622, 7718, 7730, 7928, 7982, 8432, 8444, 8558, 8660, 8828, 9008, 9020, 9122, 9290, 9308, 9422, 9722, 9860, 10964, 11012, 11090, 11498, 11972, 12062, 12098, 12548, 12602, 12878, 14018, 14162, 14882, 15758, 15908, 16172, 16838, 17168, 17648, 18428, 19142, 20330, 20918, 21548, 21722, 23018, 23612, 25022, 27668, 30212, 30668, 31130, 32162, 32372, 47702, 63758, 66410, 89072
[/CODE]

odd numbers: [URL="https://oeis.org/A346368"]A346368[/URL]

even numbers: 2*[URL="https://oeis.org/A143989"]A143989[/URL] (square number=0 allowed, prime=2 allowed)

There are many numbers which is not twice a triangular number which can be [B]uniquely[/B] written as (twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, but I conjectured there are only finite such numbers (there are 58 known such numbers, the largest such number is 16632), furthermore, for every number r>=0, there are only finitely many numbers which is not twice a triangular number which can be written as (twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, no more than r ways.

[CODE]
5, 7, 8, 11, 18, 21, 22, 24, 27, 32, 38, 50, 51, 54, 57, 60, 62, 74, 84, 105, 108, 111, 126, 140, 150, 174, 180, 186, 242, 252, 258, 270, 357, 372, 471, 492, 510, 630, 666, 690, 765, 792, 810, 1080, 1112, 1380, 1434, 1602, 1848, 1920, 2160, 3726, 4752, 5397, 5652, 7800, 12420, 16632
[/CODE]

even numbers: 2*[URL="https://oeis.org/A155190"]A155190[/URL] (triangular number=0 allowed, prime=2 allowed)

 sweety439 2021-11-09 17:36

2 Attachment(s)
"Number of ways" sequences in OEIS:

Conjecture 1:

odd numbers:

[URL="https://oeis.org/A143539"]A143539[/URL]((n+1)/2)

[URL="https://oeis.org/A046923"]A046923[/URL]((n-1)/2) (allow the square number to be 0)

even numbers:

[URL="https://oeis.org/A064272"]A064272[/URL](n/2) (allow the prime to be 2)

[URL="https://oeis.org/A002471"]A002471[/URL](n/2) (allow the prime to be 2) (allow the square number to be 0)

Conjecture 2:

odd numbers:

[URL="https://oeis.org/A144590"]A144590[/URL]((n-1)/2)

even numbers:

[URL="https://oeis.org/A132399"]A132399[/URL](n/2) (allow the prime to be 2) (allow the triangular number to be 0)

These two text files is the number of ways to write n as "(twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime" or "(twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime", the numbers n which is (for the former, n is itself twice a square number, for the latter n is itself twice a triangular number) are marked with star symbol "*", since they are not in our conjectures, many (in fact, almost all) such n cannot be written as this form, and we conjecture that all [I]other[/I] n which is enough large (>43358 for square numbers, >432 for triangular numbers) can be written as this form.

 sweety439 2021-11-09 17:49

Notes:

Conjecture 1:

* For a number which is twice a square number (2*n^2), can be written as (twice a nonzero square number) + (2*p), the only possible prime is when the nonzero square number is 2*(n-1)^2, because of algebraic factorization, and hence p=2*n-1, thus, if 2*n-1 is prime, then 2*n^2 can be [B]uniquely[/B] written as this way, and if 2*n-1 is not prime, then 2*n^2 cannot be written as this way, for the fully such numbers divided by 2 (allow the prime to be 2), see [URL="https://oeis.org/A064233"]A064233[/URL] (also see [URL="https://oeis.org/A014090"]A014090[/URL] for both allow the prime to be 2 and allow the square number to be 0)

Conjecture 2:

* For a number which is twice a triangular number (n*(n+1)), can be written as (twice a nonzero triangular number) + (2*p), the only possible prime is when the nonzero triangular number is (n-1)*n or (n-2)*(n-1), because of algebraic factorization, and hence p=n or p=2*n-1, thus, if only one of n and 2*n-1 is prime, then n*(n+1) can be [B]uniquely[/B] written as this way, and if neither n nor 2*n-1 is prime, then n*(n+1) cannot be written as this way, for the fully such numbers divided by 2 (allow the prime to be 2), see [URL="https://oeis.org/A111908"]A111908[/URL] (also see [URL="https://oeis.org/A076768"]A076768[/URL] for both allow the prime to be 2 and allow the triangular number to be 0)

Related OEIS sequences:

Conjecture 1:

2*n-1 is prime: [URL="https://oeis.org/A006254"]A006254[/URL]
2*n-1 is not prime: [URL="https://oeis.org/A104275"]A104275[/URL]
2*n-1 is composite: [URL="https://oeis.org/A053726"]A053726[/URL]

Conjecture 2:

n and 2*n-1 are both primes: [URL="https://oeis.org/A005382"]A005382[/URL]
n is prime but 2*n-1 is not prime: [URL="https://oeis.org/A307390"]A307390[/URL]
n is not prime but 2*n-1 is prime: [URL="https://oeis.org/A174166"]A174166[/URL]
n and 2*n-1 are both not primes: [URL="https://oeis.org/A138666"]A138666[/URL]

 sweety439 2021-11-16 04:00

4 Attachment(s)
Update related text files.

 sweety439 2021-11-17 03:53

These are:

* Numbers not in [URL="https://oeis.org/A001105"]A001105[/URL] which cannot be written as [URL="https://oeis.org/A001105"]A001105[/URL](i) + [URL="https://oeis.org/A085118"]A085118[/URL](j) for i >= 1, j >= 2 (note: i=0 and/or j=1 have been excluded)

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}

* Numbers not in [URL="https://oeis.org/A002378"]A002378[/URL] which cannot be written as [URL="https://oeis.org/A002378"]A002378[/URL](i) + [URL="https://oeis.org/A085118"]A085118[/URL](j) for i >= 1, j >= 2 (note: i=0 and/or j=1 have been excluded)

{1, 3, 4, 10, 14, 122, 422, 432}

 sweety439 2021-11-17 03:59

Conjecture about prime numbers and square/triangular numbers

[B]Conjecture 1[/B]: Except these 47 numbers, all numbers which is not twice a square number ([URL="https://oeis.org/A001105"]A001105[/URL]) can be written as (twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime:

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}

(i.e. except these 47 numbers, all positive integers which is not in [URL="https://oeis.org/A001105"]A001105[/URL] can be written as [URL="https://oeis.org/A001105"]A001105[/URL](i) + [URL="https://oeis.org/A085118"]A085118[/URL](j) for i >= 1, j >= 2 (note: i=0 and/or j=1 have been excluded))

[B]Conjecture 2[/B]: Except these 8 numbers, all numbers which is not twice a triangular number ([URL="https://oeis.org/A002378"]A002378[/URL]) can be written as (twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime:

{1, 3, 4, 10, 14, 122, 422, 432}

(i.e. except these 8 numbers, all positive integers which is not in [URL="https://oeis.org/A002378"]A002378[/URL] can be written as [URL="https://oeis.org/A002378"]A002378[/URL](i) + [URL="https://oeis.org/A085118"]A085118[/URL](j) for i >= 1, j >= 2 (note: i=0 and/or j=1 have been excluded))

 sweety439 2021-11-17 04:20

PARI/GP program:

[CODE]
isp(n)=(n==0)||(isprime(n)&&n>2)
is(n)=for(k=1,n,if(isp((n-2*k^2)/(2-(n%2))),return(0)));1
iss(n)=for(k=1,n,if(isp((n-k*(k+1))/(2-(n%2))),return(0)));1
[/CODE]

another PARI/GP program:

[CODE]
isoddprime(n)=isprime(n)&&n>2
a(n)=sum(k=1,n,isoddprime((n-2*k^2)/(2-(n%2))))
b(n)=sum(k=1,n,isoddprime((n-k*(k+1))/(2-(n%2))))
[/CODE]

 sweety439 2021-11-18 04:56

[QUOTE=sweety439;592782]

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}

[/QUOTE]

For these counterexamples for conjecture 1, all numbers >10 except 1542 are == 2 mod 3 (reference: [URL="https://oeis.org/A060003"]https://oeis.org/A060003[/URL], this says that these counterexamples for conjecture 1, all odd numbers >3 are == 2 mod 3)

 sweety439 2021-11-18 05:41

The counterexamples listed as sequences:

(A) Odd counterexamples
(AA) Odd prime counterexamples
(AB) Odd composite counterexamples
(AC) Odd unit (only for "1") counterexamples
[note: Odd counterexamples cannot allow the even prime 2, since (twice a square/triangular number) + (the even prime (2)) will be even number, not odd number]
(B) Even counterexamples
(BA) Even "twice prime" counterexamples
(BAA) Even "twice prime" counterexamples which are still counterexamples even if allowing the even prime 2
(BAB) Even "twice prime" counterexamples which if allowing the even prime 2, then will not be counterexamples
(BB) Even "twice composite" counterexamples
(BBA) Even "twice composite" counterexamples which are still counterexamples even if allowing the even prime 2
(BBB) Even "twice composite" counterexamples which if allowing the even prime 2, then will not be counterexamples
[note: no even "twice unit (only for "2")" counterexamples, since the number "2" is both "twice square number" and "twice triangular number", thus not in my conjectures]
[I](C) Even "not counterexamples" since they are "twice square numbers" or "twice triangular numbers", there are infinitely many such numbers[/I]
[I](CA) Even "not counterexamples" which are still counterexamples even if allowing the even prime 2[/I]
[I](CB) Even "not counterexamples" which if allowing the even prime 2, then will not be counterexamples[/I]

Conjecture 1 (positive square number) + (odd prime (or) twice odd prime):

(A) {1, 3, 17, 137, 227, 977, 1187, 1493, 5777, 5993} ([URL="https://oeis.org/A060003"]A060003[/URL])
(AA) {3, 17, 137, 227, 977, 1187, 1493} ([URL="https://oeis.org/A042978"]A042978[/URL] \ {2})
(AB) {5777, 5993}
(AA+AB) {3, 17, 137, 227, 977, 1187, 1493, 5777, 5993}
(AC) {1}
(B+C) (2*[URL="https://oeis.org/A064233"]A064233[/URL] U {6})
(BAA+BBA+CA) (2*[URL="https://oeis.org/A064233"]A064233[/URL])
(B) {4, 6, 10, 20, 26, 62, 68, 74, 116, 122, 170, 182, 254, 260, 428, 452, 740, 758, 878, 1052, 1142, 1412, 1460, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5972, 6638, 7352, 15098, 19268, 43358}
(BAA+BBA) {4, 10, 20, 26, 62, 68, 74, 116, 122, 170, 182, 254, 260, 428, 452, 740, 758, 878, 1052, 1142, 1412, 1460, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5972, 6638, 7352, 15098, 19268, 43358}
(BAB+BBB) {6}
(BA) {4, 6, 10, 26, 62, 74, 122, 254, 758, 878, 1142, 1658, 1982, 3098, 6638, 15098} (2*[URL="https://oeis.org/A065377"]A065377[/URL] U {6})
(BAA) {4, 10, 26, 62, 74, 122, 254, 758, 878, 1142, 1658, 1982, 3098, 6638, 15098} (2*[URL="https://oeis.org/A065377"]A065377[/URL])
(BAB) {6}
(BB+C) (2*[URL="https://oeis.org/A014090"]A014090[/URL] U {})
(BBA+CA) (2*[URL="https://oeis.org/A014090"]A014090[/URL])
(BB) {20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5972, 7352, 19268, 43358} (2*[URL="https://oeis.org/A020495"]A020495[/URL] U {})
(BBA) {20, 68, 116, 170, 182, 260, 428, 452, 740, 1052, 1412, 1460, 1542, 2510, 2702, 2828, 3812, 5972, 7352, 19268, 43358} (2*[URL="https://oeis.org/A020495"]A020495[/URL])
(BBB) {}
(C) (twice_square([URL="https://oeis.org/A104275"]A104275[/URL]) U {})
(CA) (twice_square([URL="https://oeis.org/A104275"]A104275[/URL]))
(CB) {}

Conjecture 2 (positive triangular number) + (odd prime (or) twice odd prime):

(A) {1, 3}
(AA) {3}
(AB) {}
(AA+AB) {3}
(AC) {1}
(B+C) (2*[URL="https://oeis.org/A111908"]A111908[/URL] U {6, 10})
(BAA+BBA+CA) (2*[URL="https://oeis.org/A111908"]A111908[/URL])
(B) {4, 10, 14, 122, 422, 432} (2*[URL="https://oeis.org/A255904"]A255904[/URL] U {10})
(BAA+BBA) {4, 14, 122, 422, 432} (2*[URL="https://oeis.org/A255904"]A255904[/URL])
(BAB+BBB) {10}
(BA) {4, 10, 14, 122, 422} (2*[URL="https://oeis.org/A065397"]A065397[/URL] U {10})
(BAA) {4, 14, 122, 422} (2*[URL="https://oeis.org/A065397"]A065397[/URL])
(BAB) {10}
(BB+C) (2*[URL="https://oeis.org/A076768"]A076768[/URL] U {6})
(BBA+CA) (2*[URL="https://oeis.org/A076768"]A076768[/URL])
(BB) {432}
(BBA) {432}
(BBB) {}
(C) (twice_triangular([URL="https://oeis.org/A138666"]A138666[/URL]) U {6})
(CA) (twice_triangular([URL="https://oeis.org/A138666"]A138666[/URL]))
(CB) {6}

 sweety439 2021-11-18 06:09

Also:

Conjecture 1:

(A) [URL="https://oeis.org/A143539"]A143539[/URL] = 0
(AB) [URL="https://oeis.org/A046923"]A046923[/URL] = 0
(BAA+BBA+CA) [URL="https://oeis.org/A064272"]A064272[/URL] = 0
(BBA+CA) [URL="https://oeis.org/A002471"]A002471[/URL] = 0

Conjecture 2:

(A) [URL="https://oeis.org/A144590"]A144590[/URL] = 0
(BBA+CA) [URL="https://oeis.org/A132399"]A132399[/URL] = 0

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