Severity: Notice
Message: Undefined offset: 0
Filename: controllers/Abstracts.php
Line Number: 95
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 95
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Trying to access array offset on value of type null
Filename: controllers/Abstracts.php
Line Number: 95
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 95
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: id
Filename: controllers/Abstracts.php
Line Number: 97
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 97
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: slug
Filename: controllers/Abstracts.php
Line Number: 100
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 100
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: short_name
Filename: controllers/Abstracts.php
Line Number: 101
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 101
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: id
Filename: controllers/Abstracts.php
Line Number: 105
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 105
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: id
Filename: controllers/Abstracts.php
Line Number: 107
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 107
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: id
Filename: controllers/Abstracts.php
Line Number: 109
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 109
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: image
Filename: controllers/Abstracts.php
Line Number: 127
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 127
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: name
Filename: controllers/Abstracts.php
Line Number: 138
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 138
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
PTZ: We're glad you're here. Please click "create a new query" if you are a new visitor to our website and need further information from us.
If you are already a member of our network and need to keep track of any developments regarding a question you have already submitted, click "take me to my Query."
Severity: Notice
Message: Undefined index: image
Filename: views/abstract.php
Line Number: 38
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 38
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: name
Filename: views/abstract.php
Line Number: 38
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 38
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: short_name
Filename: views/abstract.php
Line Number: 47
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 47
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: slug
Filename: views/abstract.php
Line Number: 48
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 48
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: slug
Filename: views/abstract.php
Line Number: 50
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 50
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: slug
Filename: views/abstract.php
Line Number: 53
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 53
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: slug
Filename: views/abstract.php
Line Number: 56
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 56
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: slug
Filename: views/abstract.php
Line Number: 273
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 273
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Severity: Notice
Message: Undefined index: name
Filename: views/abstract.php
Line Number: 273
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 273
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once
Here we analyze three well-known conjectures: (i) the existence of infinitely many twin primes, (ii) Goldbach's strong conjecture, and (iii) Polignac's conjecture. We show that the three conjectures are related to each other. In particular, we see that in analysing the validity of Goldbach's strong conjecture, one must consider also the existence of an infinite number of twin primes. As a consequence of how we approach this analysis, we also observe that if this conjecture is true, then so is Polignac's conjecture. Our first step is an analysis of the existence of infinitely many twin prime numbers. For this, using the formula 4((n−1)!+1) ≡−n (mod n(n + 2)) – satisfied if and only if (n, n + 2) are twin primes –, together with Wilson's theorem, we obtain conditions that must be met for two numbers to be twin primes. Our results, obtained from an analytic and functional study, lead us to conclude that there may exist infinitely many twin primes. Next, we consider the validity of Goldbach's strong conjecture. After showing that the conjecture is true for the first even numbers, we notice a pattern that we analyze for any even number, reducing it to three cases: (i) when the even number 2n is two times a prime number n; (ii) when the even number 2n is such that n=2m, with 2m-1 and 2m+1 twin primes; (iii) all other cases, i.e., for any 2n even number ∀n∈N with n > 1, n prime or not, with independence of n=2m being 2m-1 and 2m+1 twin primes or not. In this last case, we show that one can always find a certain r∈N such that 1 < r < n satisfying that n − r and n + r are primes, so that their sum is 2n. In this case, we use the reduction to absurd method, and our results lead us to conclude that Goldbach's strong conjecture is true to the best of our calculations, and Polignac's conjecture as well.
Keywords:
Published on: Mar 21, 2024 Pages: 76-84
Full Text PDF
Full Text HTML
DOI: 10.17352/amp.000110
CrossMark
Publons
Harvard Library HOLLIS
Search IT
Semantic Scholar
Get Citation
Base Search
Scilit
OAI-PMH
ResearchGate
Academic Microsoft
GrowKudos
Universite de Paris
UW Libraries
SJSU King Library
SJSU King Library
NUS Library
McGill
DET KGL BIBLiOTEK
JCU Discovery
Universidad De Lima
WorldCat
VU on WorldCat
Severity: Notice
Message: Undefined index: short_name
Filename: views/abstract.php
Line Number: 572
Backtrace:
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/views/abstract.php
Line: 572
Function: _error_handler
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/views/templates/site/mini-header-template.php
Line: 90
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 357
Function: include
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/third_party/MX/Loader.php
Line: 300
Function: _ci_load
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/core/MY_Controller.php
Line: 101
Function: view
File: /var/www/vhosts/peertechzpublications.org/httpdocs/application/modules/abstracts/controllers/Abstracts.php
Line: 141
Function: _render_page
File: /var/www/vhosts/peertechzpublications.org/httpdocs/index.php
Line: 315
Function: require_once