Just trying out Academia.edu with a few test upoads to link to from elsewhere. Pls excuse a few typos - haven't had time to fix them. Autodidactic IT and Electronics Engineer 4+ decades in commercial computer systems including computer forensics, BI, and big data.
Personal interest in cosmology, the physics of time, and challenging the concept of space-time as an incorrect model of time. Supervisors: Independent researcher
The author of this paper claims many simple proofs of FLT are now accessible using recent discove... more The author of this paper claims many simple proofs of FLT are now accessible using recent discoveries involving Pascal's Triangle, and in particular, Moessner's Sieve.
Naive proofs of FLT abandon the constraint of the Natural number domain for Reals, unnecessarily converting a simple proof of 'that which is' into an impossible proof of 'that which is not'. The latter requires visibility into the entire solution set in the Real domain which is infinitely more vast than limiting the domain to only Naturals.
In this revised approach from 2015, the Pythagorean is used as the starting point to define a novel factorisation of x^n = h^n - y^n using a single factor (h-y) and additional instances of (h+y) for each increase of n.
This proof by contradiction clearly shows the how the inner terms of the expansion of (h +y)^n (h - y) for all powers of n must all be zero to render FLT false, and why those conditions can only be met when n=2.
This is a Wiles independent proof using methods available to Fermat and is the first of two different simple approaches the author claims to be rigorous simple proofs of Fermat's Last Theorem.
Performing calculations involving recurrence in decimal positional notation is often problematic ... more Performing calculations involving recurrence in decimal positional notation is often problematic because recurrence implies divide by zero. Evidently it is not being taught in schools that "0.999…" (where "…" indicates a recuring last digit) being equated to 1 is by convention only, to compensate for rounding errors in calculations. Values involving recurrence are typically end results and not used for further calculation. However, as recurrence involving "0.999…" can be the result of a rounding error, by convention its value can be approximated to 1 in such cases. This is analogous to "0.556" being used as a placeholder approximation for "0.555…" to enable further calculation. This convention does not mean that an infinitesimal rounding error can be mischievously reduced by infinite descent to zero (Zeno's method) to "prove" equality. Proofs claiming that "0.999… = 1" or the like are viral mathematical jokes/hoaxes/scams circulating on the internet at the time of writing that have deceived many high profile mathematicians around the world, including mathematics professors and mathematics departments. To be clear, the construct 0.999.. is that of an irrational number not expressible as an integer division (rational).
Proofs claiming that "0.999… = 1" or the like (where "…" indicates a recuring last digit) are vir... more Proofs claiming that "0.999… = 1" or the like (where "…" indicates a recuring last digit) are viral mathematical jokes/hoaxes/scams circulating on the internet at the time of writing that have deceived many high profile mathematicians around the world, including mathematics professors and entire mathematics departments. Evidently it is not being taught that "0.999…" being equated to 1 is by convention only, in order to compensate for rounding errors in some calculations. Values involving recurrence are typically end results and not used for further calculation. However, as recurrence involving "0.999…" can be the result of a rounding error, by convention, its value can be assumed to be 1 in such cases. For example, "0.556" can be used as a placeholder or proxy for "0.555…" to enable ease of further calculation. This convention does not mean that an infinitesimal rounding error can deliberately contrived by infinite reduction to zero in order to prove the two are equal.
This is a serious proof of Fermat's Last Theorem.
It is, by inspection, quite clear that there ... more This is a serious proof of Fermat's Last Theorem.
It is, by inspection, quite clear that there is no trivial solution to the problem solved by Wiles (et. al.). This solution is to a different problem to the one solved by Wiles (et. al.) and it is trivial. It is only 1 page. It is very simple. Anyone with high school maths skills can understand it. It has been put forward by others, typically in unnecessary and voluminous detail, without acknowledging it as a different problem from that solved by Wiles (et. al.).
The difference is that, in the trivial solution, the Pythagorean is assumed and it is simply a matter of substitution to show FLT is true in the context of the Pythagorean. This trivial proof addresses the mostly ignored case of Fermat mistaking his proof as "Marvelous" until he realised the problem itself, although original, was completely trivial. It is only when one is prepared to entertain this case, that the much sought after "Marvelous" proof appears for what it is - a trivial proof of a trivial problem.
I have no doubt this is what Fermat had in mind. FLT has been incorrectly understood as a far more complex problem that was actually stated because of ambiguity and omission in Fermat's description.
Prof Alain Aspect's 2011 lecture 1 delivered to the Technion Physics Colloquium on June 13, 2011,... more Prof Alain Aspect's 2011 lecture 1 delivered to the Technion Physics Colloquium on June 13, 2011, clearly explains the Double Slit Delayed Choice Experiment devised by Wheeler and implemented by Aspect using polarising filters as so called " beam splitters ". This paper asserts that the effect of circular polarisation has been overlooked in this famous and ingenious experiment, and that the unexpected interference patterns are merely a result of circular polarisation, not of time reversal, nor of effect predetermining cause, nor any quantum " weirdness ". It is further asserted that linear polarisation is also at work in Thomas Young's original Double Slit experiment, but these circular polarisation effects do not affect the " interference " patterns in Young's experiment. DOUBLE SLIT RESULTS EXPLAINED Circular polarisation is the phenomenon used in 3D Television 2. Circular polarisation is a necessary consideration in Alain Aspect's apparatus used to perform Wheelers Double Slit Delayed Choice experiment. However circular polarisation has not been discussed by Aspect, let alone as potentially causative of the unexpected interference pattern he observes. This omission is noteworthy because the explanation offered by Aspect in lieu of circular polarisation effects is far more astonishing even than the not insignificant claim in this paper proposing a far more prosaic cause. The consequence of circular polarisation is that the interference pattern-the inexplicable " weird " result-is nothing more than the effect of polarisation. The linear polarising beam splitters modulate circular polarisation inherent in the photon to randomly modulate each photon thus producing an apparent interference effect. (This occurs regardless of the successful use of Fock states to actually fire off just "one photon" at a time, claimed by Aspect to be significant). But features of Young's interference pattern are missing-most notably-there is no ability to alter the size of the "slits" which are non-existent in Aspects apparatus having been replaced by polarising "beam splitters". Thus we cannot account for the distance between the peaks of the interference pattern by adjusting the gaps and observing its direct effect on the peak to peak distance (etc). Neither can we say what the size and separation of the double slits are if there are none. No analysis has been attempted on the interference pattern to trace a root cause. Yet, without knowledge of the apparatus, what we see at face value is a typical interference pattern observed in Young's double slit experiment. In Young's experiment linear polarisation is induced in the light beam by the double slit as a somewhat unintended consequence of the experiment. The effect of this polarisation, would if anything, enhance the sharpness of the interference pattern. The effect of circular polarisation in
ABSTRACT
This is a proof to show that 0.9999... is not = 1 (where '...' means recurring)1. This i... more ABSTRACT This is a proof to show that 0.9999... is not = 1 (where '...' means recurring)1. This is all that is necessary to refute the class of problems known generally as Zeno’s Paradox2.
There are only two parts to this proof.
Firstly, I represent 1 as 1.000... and compare the finite decimal representation of 1.000...0 with its finite counterpart 0.999...9 (where the rightmost digit indicates a finite recurrence to an equivalent arbitrary accuracy) and show they can never be equal if the recurring digit recurs finitely.
Then I show that the number system breaks down if the recurring digit recurs infinitely because the interval between all numbers becomes zero and a Limit at infinity cannot and does not exist.
I conclude that recurring digits and asserting 0.999… = 1 (etc) are a mathematically incorrect non sequiturs arising from a failure to acknowledge a limitation of the decimal notation number system.
The author of this paper claims many simple proofs of FLT are now accessible using recent discove... more The author of this paper claims many simple proofs of FLT are now accessible using recent discoveries involving Pascal's Triangle, and in particular, Moessner's Sieve.
Naive proofs of FLT abandon the constraint of the Natural number domain for Reals, unnecessarily converting a simple proof of 'that which is' into an impossible proof of 'that which is not'. The latter requires visibility into the entire solution set in the Real domain which is infinitely more vast than limiting the domain to only Naturals.
In this revised approach from 2015, the Pythagorean is used as the starting point to define a novel factorisation of x^n = h^n - y^n using a single factor (h-y) and additional instances of (h+y) for each increase of n.
This proof by contradiction clearly shows the how the inner terms of the expansion of (h +y)^n (h - y) for all powers of n must all be zero to render FLT false, and why those conditions can only be met when n=2.
This is a Wiles independent proof using methods available to Fermat and is the first of two different simple approaches the author claims to be rigorous simple proofs of Fermat's Last Theorem.
Performing calculations involving recurrence in decimal positional notation is often problematic ... more Performing calculations involving recurrence in decimal positional notation is often problematic because recurrence implies divide by zero. Evidently it is not being taught in schools that "0.999…" (where "…" indicates a recuring last digit) being equated to 1 is by convention only, to compensate for rounding errors in calculations. Values involving recurrence are typically end results and not used for further calculation. However, as recurrence involving "0.999…" can be the result of a rounding error, by convention its value can be approximated to 1 in such cases. This is analogous to "0.556" being used as a placeholder approximation for "0.555…" to enable further calculation. This convention does not mean that an infinitesimal rounding error can be mischievously reduced by infinite descent to zero (Zeno's method) to "prove" equality. Proofs claiming that "0.999… = 1" or the like are viral mathematical jokes/hoaxes/scams circulating on the internet at the time of writing that have deceived many high profile mathematicians around the world, including mathematics professors and mathematics departments. To be clear, the construct 0.999.. is that of an irrational number not expressible as an integer division (rational).
Proofs claiming that "0.999… = 1" or the like (where "…" indicates a recuring last digit) are vir... more Proofs claiming that "0.999… = 1" or the like (where "…" indicates a recuring last digit) are viral mathematical jokes/hoaxes/scams circulating on the internet at the time of writing that have deceived many high profile mathematicians around the world, including mathematics professors and entire mathematics departments. Evidently it is not being taught that "0.999…" being equated to 1 is by convention only, in order to compensate for rounding errors in some calculations. Values involving recurrence are typically end results and not used for further calculation. However, as recurrence involving "0.999…" can be the result of a rounding error, by convention, its value can be assumed to be 1 in such cases. For example, "0.556" can be used as a placeholder or proxy for "0.555…" to enable ease of further calculation. This convention does not mean that an infinitesimal rounding error can deliberately contrived by infinite reduction to zero in order to prove the two are equal.
This is a serious proof of Fermat's Last Theorem.
It is, by inspection, quite clear that there ... more This is a serious proof of Fermat's Last Theorem.
It is, by inspection, quite clear that there is no trivial solution to the problem solved by Wiles (et. al.). This solution is to a different problem to the one solved by Wiles (et. al.) and it is trivial. It is only 1 page. It is very simple. Anyone with high school maths skills can understand it. It has been put forward by others, typically in unnecessary and voluminous detail, without acknowledging it as a different problem from that solved by Wiles (et. al.).
The difference is that, in the trivial solution, the Pythagorean is assumed and it is simply a matter of substitution to show FLT is true in the context of the Pythagorean. This trivial proof addresses the mostly ignored case of Fermat mistaking his proof as "Marvelous" until he realised the problem itself, although original, was completely trivial. It is only when one is prepared to entertain this case, that the much sought after "Marvelous" proof appears for what it is - a trivial proof of a trivial problem.
I have no doubt this is what Fermat had in mind. FLT has been incorrectly understood as a far more complex problem that was actually stated because of ambiguity and omission in Fermat's description.
Prof Alain Aspect's 2011 lecture 1 delivered to the Technion Physics Colloquium on June 13, 2011,... more Prof Alain Aspect's 2011 lecture 1 delivered to the Technion Physics Colloquium on June 13, 2011, clearly explains the Double Slit Delayed Choice Experiment devised by Wheeler and implemented by Aspect using polarising filters as so called " beam splitters ". This paper asserts that the effect of circular polarisation has been overlooked in this famous and ingenious experiment, and that the unexpected interference patterns are merely a result of circular polarisation, not of time reversal, nor of effect predetermining cause, nor any quantum " weirdness ". It is further asserted that linear polarisation is also at work in Thomas Young's original Double Slit experiment, but these circular polarisation effects do not affect the " interference " patterns in Young's experiment. DOUBLE SLIT RESULTS EXPLAINED Circular polarisation is the phenomenon used in 3D Television 2. Circular polarisation is a necessary consideration in Alain Aspect's apparatus used to perform Wheelers Double Slit Delayed Choice experiment. However circular polarisation has not been discussed by Aspect, let alone as potentially causative of the unexpected interference pattern he observes. This omission is noteworthy because the explanation offered by Aspect in lieu of circular polarisation effects is far more astonishing even than the not insignificant claim in this paper proposing a far more prosaic cause. The consequence of circular polarisation is that the interference pattern-the inexplicable " weird " result-is nothing more than the effect of polarisation. The linear polarising beam splitters modulate circular polarisation inherent in the photon to randomly modulate each photon thus producing an apparent interference effect. (This occurs regardless of the successful use of Fock states to actually fire off just "one photon" at a time, claimed by Aspect to be significant). But features of Young's interference pattern are missing-most notably-there is no ability to alter the size of the "slits" which are non-existent in Aspects apparatus having been replaced by polarising "beam splitters". Thus we cannot account for the distance between the peaks of the interference pattern by adjusting the gaps and observing its direct effect on the peak to peak distance (etc). Neither can we say what the size and separation of the double slits are if there are none. No analysis has been attempted on the interference pattern to trace a root cause. Yet, without knowledge of the apparatus, what we see at face value is a typical interference pattern observed in Young's double slit experiment. In Young's experiment linear polarisation is induced in the light beam by the double slit as a somewhat unintended consequence of the experiment. The effect of this polarisation, would if anything, enhance the sharpness of the interference pattern. The effect of circular polarisation in
ABSTRACT
This is a proof to show that 0.9999... is not = 1 (where '...' means recurring)1. This i... more ABSTRACT This is a proof to show that 0.9999... is not = 1 (where '...' means recurring)1. This is all that is necessary to refute the class of problems known generally as Zeno’s Paradox2.
There are only two parts to this proof.
Firstly, I represent 1 as 1.000... and compare the finite decimal representation of 1.000...0 with its finite counterpart 0.999...9 (where the rightmost digit indicates a finite recurrence to an equivalent arbitrary accuracy) and show they can never be equal if the recurring digit recurs finitely.
Then I show that the number system breaks down if the recurring digit recurs infinitely because the interval between all numbers becomes zero and a Limit at infinity cannot and does not exist.
I conclude that recurring digits and asserting 0.999… = 1 (etc) are a mathematically incorrect non sequiturs arising from a failure to acknowledge a limitation of the decimal notation number system.
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Papers by Marcus Anderson
Naive proofs of FLT abandon the constraint of the Natural number domain for Reals, unnecessarily converting a simple proof of 'that which is' into an impossible proof of 'that which is not'. The latter requires visibility into the entire solution set in the Real domain which is infinitely more vast than limiting the domain to only Naturals.
In this revised approach from 2015, the Pythagorean is used as the starting point to define a novel factorisation of x^n = h^n - y^n using a single factor (h-y) and additional instances of (h+y) for each increase of n.
This proof by contradiction clearly shows the how the inner terms of the expansion of (h +y)^n (h - y) for all powers of n must all be zero to render FLT false, and why those conditions can only be met when n=2.
This is a Wiles independent proof using methods available to Fermat and is the first of two different simple approaches the author claims to be rigorous simple proofs of Fermat's Last Theorem.
It is, by inspection, quite clear that there is no trivial solution to the problem solved by Wiles (et. al.). This solution is to a different problem to the one solved by Wiles (et. al.) and it is trivial. It is only 1 page. It is very simple. Anyone with high school maths skills can understand it. It has been put forward by others, typically in unnecessary and voluminous detail, without acknowledging it as a different problem from that solved by Wiles (et. al.).
The difference is that, in the trivial solution, the Pythagorean is assumed and it is simply a matter of substitution to show FLT is true in the context of the Pythagorean. This trivial proof addresses the mostly ignored case of Fermat mistaking his proof as "Marvelous" until he realised the problem itself, although original, was completely trivial. It is only when one is prepared to entertain this case, that the much sought after "Marvelous" proof appears for what it is - a trivial proof of a trivial problem.
I have no doubt this is what Fermat had in mind. FLT has been incorrectly understood as a far more complex problem that was actually stated because of ambiguity and omission in Fermat's description.
This is a proof to show that 0.9999... is not = 1 (where '...' means recurring)1. This is all that is necessary to refute the class of problems known generally as Zeno’s Paradox2.
There are only two parts to this proof.
Firstly, I represent 1 as 1.000... and compare the finite decimal representation of 1.000...0 with its finite counterpart 0.999...9 (where the rightmost digit indicates a finite recurrence to an equivalent arbitrary accuracy) and show they can never be equal if the recurring digit recurs finitely.
Then I show that the number system breaks down if the recurring digit recurs infinitely because the interval between all numbers becomes zero and a Limit at infinity cannot and does not exist.
I conclude that recurring digits and asserting 0.999… = 1 (etc) are a mathematically incorrect non sequiturs arising from a failure to acknowledge a limitation of the decimal notation number system.
Naive proofs of FLT abandon the constraint of the Natural number domain for Reals, unnecessarily converting a simple proof of 'that which is' into an impossible proof of 'that which is not'. The latter requires visibility into the entire solution set in the Real domain which is infinitely more vast than limiting the domain to only Naturals.
In this revised approach from 2015, the Pythagorean is used as the starting point to define a novel factorisation of x^n = h^n - y^n using a single factor (h-y) and additional instances of (h+y) for each increase of n.
This proof by contradiction clearly shows the how the inner terms of the expansion of (h +y)^n (h - y) for all powers of n must all be zero to render FLT false, and why those conditions can only be met when n=2.
This is a Wiles independent proof using methods available to Fermat and is the first of two different simple approaches the author claims to be rigorous simple proofs of Fermat's Last Theorem.
It is, by inspection, quite clear that there is no trivial solution to the problem solved by Wiles (et. al.). This solution is to a different problem to the one solved by Wiles (et. al.) and it is trivial. It is only 1 page. It is very simple. Anyone with high school maths skills can understand it. It has been put forward by others, typically in unnecessary and voluminous detail, without acknowledging it as a different problem from that solved by Wiles (et. al.).
The difference is that, in the trivial solution, the Pythagorean is assumed and it is simply a matter of substitution to show FLT is true in the context of the Pythagorean. This trivial proof addresses the mostly ignored case of Fermat mistaking his proof as "Marvelous" until he realised the problem itself, although original, was completely trivial. It is only when one is prepared to entertain this case, that the much sought after "Marvelous" proof appears for what it is - a trivial proof of a trivial problem.
I have no doubt this is what Fermat had in mind. FLT has been incorrectly understood as a far more complex problem that was actually stated because of ambiguity and omission in Fermat's description.
This is a proof to show that 0.9999... is not = 1 (where '...' means recurring)1. This is all that is necessary to refute the class of problems known generally as Zeno’s Paradox2.
There are only two parts to this proof.
Firstly, I represent 1 as 1.000... and compare the finite decimal representation of 1.000...0 with its finite counterpart 0.999...9 (where the rightmost digit indicates a finite recurrence to an equivalent arbitrary accuracy) and show they can never be equal if the recurring digit recurs finitely.
Then I show that the number system breaks down if the recurring digit recurs infinitely because the interval between all numbers becomes zero and a Limit at infinity cannot and does not exist.
I conclude that recurring digits and asserting 0.999… = 1 (etc) are a mathematically incorrect non sequiturs arising from a failure to acknowledge a limitation of the decimal notation number system.